For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

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### Calculating Limits as x Goes to Infinity

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Limit as x Goes to Infinity 0:14
- Limit as x Goes to Infinity
- Let's Look at f(x) = 1 / (x-3)
- Summary
- Example I: Calculating Limits as x Goes to Infinity 12:16
- Example II: Calculating Limits as x Goes to Infinity 21:22
- Example III: Calculating Limits as x Goes to Infinity 24:10
- Example IV: Calculating Limits as x Goes to Infinity 36:00

### AP Calculus AB Online Prep Course

### Transcription: Calculating Limits as x Goes to Infinity

*Hello, welcome back to www.educator.com, and welcome back to AP Calculus.*0000

*Today, we are going to talk about what happens when x actually goes to infinity.*0004

*Calculating limits, the way that we have done for previous problems.*0009

*Let us jump right on in.*0013

*We have already seen through some examples,*0016

*I think I’m going to work in red today.*0024

*I think I might work in blue.*0030

*We have already seen that the following is possible.*0033

*We have the limit as x approaches some number a of f(x) = positive or negative infinity.*0045

*We know that as we get close to some number, no matter what that number is,*0054

*that the function itself actually just blows up to infinity.*0058

*They go straight up or straight down to negative infinity.*0061

*We have seen that already.*0064

*For example, the limit as x approaches 3 from above, I will just leave it as 3 from below.*0064

*It does not matter, of 1/ x - 3 = positive infinity.*0078

*Let us go ahead and draw this out.*0087

*That is going to be something like this.*0091

*We know that 3 itself is not in the domain because if it were 3, it is not defined.*0094

*We know that we have a vertical asymptote at 3.*0100

*Basically what happened is, as we approach 3 from above of this function, as we approach 3,*0105

*this is 3 right here, from above the function just blows up to positive infinity.*0112

*We also know that the limit as x approaches 3 from below of this function 1/ x – 3.*0119

*As we approach 3 from below, coming that way, it is going to end up going to negative infinity.*0127

*That is all, that is all this means.*0136

*As x approaches some number, the function itself goes to infinity.*0140

*We have also mentioned the following.*0146

*The limit as x approaches infinity of f(x), in other words what happens when x itself goes to infinity.*0159

*We can take a look at this example again.*0169

*This 1/ x – 3, we see that as x gets really big, the function itself just gets closer and closer to 0 from above.*0171

*The same thing here, as we go to negative infinity, the function just gets closer and closer to the x axis.*0179

*In other words, to y = 0.*0186

*That is something that we have also seen.*0188

*We used graphs and tables of values to see what happens.*0193

*Just like in the last example, we just look at the graph to see what happens to f(x), *0208

*when x goes to positive or negative infinity.*0220

*Now we want deal with this analytically.*0224

*How do we actually do it mathematically?*0227

*We want to deal with these analytically.*0235

*Let us look at the example above.*0246

*Let us look at f(x) = 1/ x – 3.*0254

*We ask ourselves, what is the limit as x approaches infinity.*0261

*Let us go ahead and take positive infinity of the function 1/ x – 3.*0273

*We already saw what happens.*0279

*The way you handle it analytically is essentially going to be the same thing.*0282

*You are basically just going to put infinity in to the function and see what happens to the function.*0286

*That is what you are asking yourself.*0291

*As x gets really big, what does the whole function do?*0294

*You essentially do the same thing as before.*0301

*Just basically plugging in, instead of plugging in a number, you are plugging in infinity.*0307

*You essentially do the same thing as before.*0315

*You plug in, I mean technically, infinity it is not a number.*0323

*You cannot really plug it in.*0327

*But again, the degree of precision here is not that big a deal, we know what is happening.*0329

*As long as we know what is happening, we do not have to worry about pedantic little issues like this.*0337

*Basically, again, just plug in infinity and ask what does the function do.*0342

*Let me see, should I do the same thing as before?*0350

*Plugging in just means take x very large in the positive or negative direction and ask how the function behaves.*0352

*Let us do it this way, for the limit as x approaches positive infinity of 1/ x – 3.*0397

*When we put in infinity here, as x goes to positive infinity, the denominator gets really big.*0410

*The denominator gets very large.*0427

*Now as the denominator gets very large, what about the function itself, 1/ the denominator.*0431

*As the denominator gets large, the function itself gets really small.*0438

*Because the denominator is a bigger and bigger and bigger number.*0441

*1/ 1000, 1/ 100000, 1/10000000, it is going to 0.*0444

*As the denominator gets huge, 1/ x - 3 gets very small.*0448

*It approaches 0, that is what is happening.*0469

*That is all what we have been doing, the same thing.*0474

*You are essentially just going to plug it in and see what happens.*0477

*If you get an answer, you are good.*0481

*If you get something that does not make sense, you are going to have to do the same as before.*0482

*You are going to have to basically manipulate the expression and then take the limit again*0486

*as x goes to positive or negative infinity, whichever they are asking for.*0489

*That is the basic process over and over again.*0492

*We say the limit as x approaches positive infinity of 1/ x - 3 = 0.*0501

*The same thing with negative infinity.*0511

*What about the limit as x approaches negative infinity of 1/ x – 3.*0519

*The same thing, if we put a negative infinity in here, it is going to be a big number in the dominator.*0524

*It is going to be a very small function.*0529

*Essentially, it is going to go to 0.*0532

*We say it approaches 0, it just approaches it from a different point.*0535

*Negative infinity, these numbers in the denominator are going to be negative.*0539

*A positive number 1 divided by a negative number, it is going to be a negative number.*0543

*It is going to approach infinity from below.*0546

*Up here, as you go towards infinity, it approaches infinity from above.*0550

*That is the only difference, that vs. that.*0554

*They both approach 0 but from opposite ends, from the top and from the bottom.*0560

*Nothing strange, pretty intuitive stuff here.*0573

*We essentially do the same thing.*0577

*Sorry, I keep repeating myself.*0580

*We essentially do the same thing.*0587

*You plug in positive or negative infinity and ask what happens.*0588

*Again, when we say plug in positive or negative infinity, we are saying as infinity gets really big,*0600

*what does the function do.*0605

*If you get something, we essentially do the same thing, as what happens.*0609

*If we get something that makes sense like a number, an actual limit, that something approaches, we can stop, that is our answer.*0627

*We can stop, this is our answer.*0643

*If we get something that does not make sense, then we manipulate the expression and take the limit again.*0652

*Manipulate the expression and take that limit again.*0683

*The same limit, the limit as x goes to positive or negative infinity.*0693

*I wonder if I should write down what i have here next.*0719

*You know what, let us just launch right into the examples.*0723

*I think that is probably the best approach.*0729

*Let me go ahead and do example here.*0734

*We want to know what the limit is, as x approaches infinity of this rational function 7x² - 3x + 14/ 5x² + 2x + 6.*0740

*As x goes to infinity, the numerator goes to infinity, the denominator goes to infinity.*0761

*You are going to end up with this thing, infinity/ infinity.*0766

*This does not make sense, we are going to have to do something to this expression.*0769

*When you see 0/0, infinity/ infinity, infinity – infinity, because we actually do not know the rates at which these two go to infinity, *0775

*we do not if it actually converges to a limit or not.*0783

*That is why it is indeterminate.*0789

*You might be thinking to yourself, this one goes to infinity, this one goes to infinity, is not just 1?*0790

*No, it is not 1 because we do not know the rate at which these go to infinity.*0794

*It is indeterminate.*0802

*Here is how you handle a rational function.*0803

*For rational functions which this is, rational functions, the manipulation is the same.*0806

*We manipulate by dividing both the numerator and denominator by the highest power of x in the denominator.*0820

*In this case, the denominator, the highest power is x².*0854

*We are going to multiply top and bottom by x².*0857

*Here it is x².*0863

*What we get is the following.*0869

*We get the limit as x approaches infinity.*0870

*When we divide the top by x², we get 7 +,*0875

*Let me write up the whole thing.*0884

*7x² - 3x + 14/ x² divided by 5x² + 2x + 6/ x².*0889

*I have not changed the expression, I just multiplied the top and bottom by 1/ x².*0902

*This becomes the limit as x goes to infinity of 7 + 3/ x + 14x²/ 5 + 2/ x + 6/ x².*0906

*I just divided the x² into everything.*0932

*And now I take the limit again, as x goes to infinity.*0935

*Now I have an x in the denominator and a number on top.*0938

*As x goes to infinity, this goes to 0, this goes to 0, this goes to 0, this goes to 0.*0941

*I’m just left with 7/5, that is my limit.*0948

*That is how you do it.*0952

*Whenever you are dealing with a rational function, just divide top and bottom by the highest power in the denominator.*0954

*And then, take the limit again and that should give you your answer.*0959

*You can also have done the following with rational functions.*0966

*As x gets really big, whether positive or negative, basically,*0971

*the term that is going to dominate is going to be the highest powers in their respective polynomials.*0974

*In the top, the term that is going to dominate is 7x² term.*0979

*These basically drop out.*0984

*They do not contribute much.*0985

*In the bottom, the 5x² + 2x + 6, this term is going to dominate.*0988

*It drops out.*0992

*What you are left with is, the limit as x goes to infinity of 7x²/ 5x² which is the limit as x goes to infinity of 7/5.*0994

*We know that the limit of the constant is just the constant itself.*1008

*You can do it that way as well, absolutely fine.*1011

*Just ignore the lower degree of x and just worry about the highest degree x, and take it from there.*1014

*That is an alternate way of doing this.*1023

*Either one is fine, whichever works for you.*1026

*Let us go ahead and state the following.*1030

*When the limit as x goes to positive or negative infinity.*1041

*When you take a limit as x goes to infinity and you actually get a finite number that we got, like we got here, the 7/5, a number,*1047

*when a limit as x goes to infinity is an actual number, this number is a horizontal asymptote.*1055

*This number is a horizontal asymptote.*1066

*It is a horizontal, just like when we have rational functions like 1/ x – 3, the denominator,*1075

*the 3 is going to be a vertical asymptote because it is not defined at 3.*1080

*3 – 3 gives us 0.*1084

*When we take limits to infinity, if those limits actually give us finite numbers, those are horizontal asymptotes.*1086

*Let us take a look at what this particular function looks like.*1094

*It looks like this.*1098

*We see that as x goes to infinity, positive infinity or negative infinity, does not matter.*1099

*Let us just do it over here.*1107

*As x approaches positive infinity, this is the graph of the function.*1114

*As x gets bigger, we see that the function is actually approaching the horizontal asymptote*1119

*which is the dashed line, which is the 7/5.*1126

*This line is y = 7/5 and this of course is y = f(x), that rational function that we just worked out.*1129

*As x goes to positive infinity, f(x) goes to 7/5.*1138

*The same thing over here.*1146

*As x goes to negative infinity, f(x) actually approaches the same value 7/5.*1148

*That is all that is happening here.*1156

*A couple of things to notice, notice that the function drops below, crosses the asymptote,*1162

*and then comes up and approaches 0.*1178

*That is not a problem.*1181

*When we take limits to infinity, we are not concerned about what happens in the center of the graph.*1182

*We are only concerned about what happens as x gets really big, positive or negative.*1187

*It is not a problem for a function to actually cross a horizontal asymptote.*1192

*It can actually cross as many times as it likes.*1198

*When you deal with trigonometric functions, you might see trigonometric and exponential functions.*1201

*You might see that it actually crosses several times *1208

*but it actually gets closer and closer to some actual number to a horizontal asymptote.*1211

*A horizontal asymptote exists, a limit exists, but it is okay is it crosses it.*1218

*We are asking what happens to the function, as we get bigger and bigger.*1223

*In other words, is it approaching some number.*1226

*Do not worry about crossing the asymptote.*1228

*Let us write that down.*1236

*Crossing a horizontal asymptote is not a problem, as long as f(x) approaches some number closer and closer.*1237

*There you go, now we have the graphical, we have the tabular, if you need that.*1270

*Now we have the analytical.*1276

*Let us do another example here.*1281

*The limit as x goes to infinity 3x³ - 2x² + 4x + 14.*1285

*This one is easy, straight polynomial.*1302

*This one is easy, as x goes to positive infinity, f(x) goes to positive infinity.*1310

*Let us do it this way.*1325

*Let us deal with the highest degree.*1327

*As x gets really big and big, the only term that is going to dominate is going to be this term, the 3x³.*1330

*We do not have to worry about those.*1335

*As x goes to positive infinity, a positive number cubed is a positive number.*1338

*F(x) is going to go to positive infinity.*1342

*As x goes to negative infinity, this is going to dominate.*1350

*A negative number cubed is a negative number.*1357

*Therefore, f(x) itself is going to end up going towards negative infinity.*1359

*That is all that is going on here.*1365

*Notice, as x goes to positive infinity, f(x) goes to positive infinity.*1376

*As x goes to negative infinity, f(x) goes to negative infinity.*1381

*This does not actually converge to a number.*1385

*The functions just fly off in opposite directions.*1387

*This function does not converge to an actual number.*1390

*That can happen, we know that already.*1407

*We have seen limit as x approaches some number can be some number, an actual limit.*1413

*The limit as x approaches some number, the function can go off to infinity.*1419

*Now x itself can go off to positive or negative infinity.*1425

*The function itself can approach some number, an actual limit.*1428

*The limit exists, in other words.*1433

*Or as x goes to positive or negative infinity, the function goes to positive or negative infinity.*1434

*Those are the possibilities.*1439

*Let us do another example.*1447

*This is going to be slightly more involved and it is definitely an example that you want to pay close attention to.*1449

*The limit as x approaches infinity of 15x² + 30 all under the radical sign/ x - 1.*1456

*We have a rational function.*1470

*There is a square root on the top but it is still a rational function and some function on top/ some function on the bottom.*1472

*Again, a rational function, our general procedure for dealing with a rational function,*1485

*it implies that we divide the top and bottom by the highest degree of x in the denominator.*1494

*We divide the numerator and denominator by the highest power of x in the denominator.*1501

*Now x going to infinity means x goes to positive infinity and x goes to negative infinity.*1528

*I have to let you know something.*1539

*Some people, when you see x goes to infinity, they are saying goes to positive infinity.*1541

*Remember what we said in the previous lesson.*1547

*When you are dealing with limits as x goes to infinity, we deal with those separately.*1548

*x = positive infinity, x = negative infinity.*1552

*Here in some books and in some classes, when people say x goes to infinity, they mean positive infinity.*1555

*They do not mean the negative.*1563

*In this case, when I write x = goes to infinity, I will specify whether I mean just positive infinity, or in this case,*1564

*it means break it up into both its positive and negative infinity.*1571

*Beware of that distinction.*1575

*We do both.*1582

*Let us deal with x being greater than 0, in other words, x going to positive infinity.*1587

*x is greater than 0.*1593

*Like we said, we divide the top and bottom by the highest power.*1596

*We are going to get 15x² + 30 under the radical/ x/ x - 1/ x.*1600

*This is going to equal, x, I can think of it as √x².*1611

*15x² + 30, under the radical/ √x².*1618

*I’m just rewriting x as the √x².*1625

*All over here, I divide 1 - 1/ x.*1629

*Since this is a radical and this is a radical, I combine the radicals.*1634

*This is going to end up equaling 15x² + 30/ x² under the radical, /1 - 1/ x.*1638

*I do the actual division, now that I’m under one radical.*1651

*I get √15 + 30/ x²/ 1 – 1/ x.*1653

*This is my particular manipulation.*1664

*Now that I have a manipulation, now I take the limit.*1667

*We take the limit as x approaches positive infinity.*1676

*The limit as x approaches positive infinity of 15 + 30/ x² under the radical, / 1 - 1/ x.*1686

*As x goes to infinity, 30/ x² goes to 0 because x² blows up.*1698

*The denominator, it is only a constant on top, it is going to go to 0.*1704

*1/ x as x goes to infinity goes to 0.*1708

*We are left with √15.*1711

*√15 is our limit as x goes to positive infinity.*1714

*Let us deal with x less than 0.*1719

*X goes to negative infinity.*1721

*For x less than 0, it gets a little strange.*1726

*We still have the same thing.*1737

*We still have the 15x² + 30/ x, same thing, / x - 1 divided by x, dividing by the highest power of x in the denominator.*1738

*We are going to do the same thing that we did before.*1751

*Except, we are going to have 15x² + 30/ √x²/ x - 1/ x.*1754

*This is exactly the same thing that we did before, except for one thing, now because x is less than 0.*1766

*This is x and we turn it into x² as a manipulation.*1775

*Because x is √x², we put a negative sign here.*1779

*The reason we do that is the following.*1783

*The reason for this negative sign, I will say notice this negative sign which was not there, when we took x positive.*1786

*Here is why.*1801

*The √x², it does not actually equal x.*1809

*The square root of x² actually equals, we know this from algebra and pre-calculus.*1814

*But we do not deal with it that much, that is the problem.*1821

*The √x² actually equals the absolute value of x.*1823

*The absolute value of x is equal to x, when x is greater than 0.*1828

*Or it equals negative x, when x is less than 0.*1835

*This negative sign has to be brought in, when you are dealing with x less than 0.*1839

*When you turn this x into a √x², this is actually an absolute value of x.*1844

*That absolute value of x is a –x, that comes out here.*1851

*That is what is going on here.*1855

*I hope that makes sense.*1861

*It is a little strange, probably you have to think about it for a little bit but that is what is happening.*1863

*The reason being that the √x² is not just x.*1867

*It is actually the absolute value of x, because when x is negative, you can square it.*1876

*You can still get a positive number that you can take the square root of.*1881

*You have to account for x being negative or x being positive.*1884

*That is why this negative sign have to show up, when we are dealing with x *1888

*which is less than 0 because of the definition of the absolute value.*1892

*I hope that makes sense, in any case.*1896

*Now we have that expression.*1899

*We have -15x² + 30 under the radical, / √x²/ x - 1/ x = -15x² + 30/ x² / 1 – 1/ x.*1902

*All of that = -√15 - 30/ x² / 1 - 1/ x.*1929

*Now we take the limit.*1940

*The limit as x approaches -infinity of -15 -,*1943

*Was it – or +, I think it is + actually + 30/ x² / 1 - 1/ x.*1953

*As x goes to negative infinity, 30/ x² goes to 0, 1/ x goes to 0.*1960

*We are left with -√15 is the limit.*1968

*Now when we took x to positive infinity, our horizontal asymptote was √15.*1975

*As x goes to negative infinity, our horizontal asymptote is -√15.*1984

*You have two horizontal asymptotes.*1989

*The function is behaving differently.*1991

*It is approaching two different numbers, as you go big to the right and big to the left.*1993

*The function f(x) = the original function is 30/ x - 1 has two horizontal asymptotes. *2006

*y = √15 and y = -√15.*2030

*Let us take a look and see what this actually looks like.*2038

*It is going to look like this.*2042

*This is our function and the dashed lines are going to be the horizontal and vertical asymptotes.*2043

*This line right here, this is your y = √15.*2051

*This line right here is y = -√15.*2057

*I hope that makes sense.*2064

*Notice we also happen to have a vertical asymptote because of the x – 1 in the denominator.*2067

*Here, this is our vertical asymptote.*2072

*But as the function, here is the 0,0 mark right here.*2080

*As x gets really big, the function, as you can see, gets closer and closer and closer to √15.*2085

*As x gets really big, negative goes to negative infinity.*2094

*The function crosses and then comes back down, and gets closer and closer and closer and closer to -√15.*2099

*Whenever you see radicals in rational functions, things like this are going to happen.*2108

*You just have to be careful.*2113

*Again, let us recall that √x² actually = the absolute value of x.*2114

*The absolute value of x is equal to regular x, when x is greater than 0.*2121

*In other words, when you are going to positive infinity but it is equal to –x, when x is less than 0.*2126

*When you are going to negative infinity.*2132

*You have to have that extra negative sign.*2134

*It is going to confuse the heck out of you and do not worry about it.*2137

*To this day, I still make mistakes with stuff like this.*2140

*Which is why personally, I love graphs.*2144

*I make the graph and the graph tells me exactly what my function is doing.*2147

*And then, I adjust my analysis in order to fit the graph.*2150

*It is cheating but c'est la vie.*2155

*Let us do another, an example.*2160

*What is the limit as x approaches 0 of e¹/x.*2167

*This is a number, x = 0.*2181

*We have to do x approaches 0 from above.*2183

*We have to do x approaches 0 from below.*2191

*For x approaching 0 from above, positive numbers headed towards 0, here is what we get.*2197

*As x gets close to 0, this 1/x goes to positive infinity.*2208

*As x gets tinier and tinier, 1/ 1/10, 1/ 1/100, 1/ 1/1000000, the 1/x blows up to infinity.*2220

*We get e ⁺infinity = infinity.*2232

*The limit as x approaches 0 from above of e¹/x = positive infinity.*2240

*For x approaches 0 from below, as x approaches 0 from below, *2253

*x is a negative number that means 1/x goes to negative infinity.*2269

*x is a negative number.*2281

*It is getting closer to 0 but it is still a negative number.*2283

*1/x is a negative number.*2288

*It is going to end up going towards negative infinity.*2290

*1/x goes to negative infinity.*2293

*e ⁺negative infinity is equivalent to 1/ e ⁺infinity.*2298

*1/e, as this becomes infinitely large, the 1/x becomes infinitely small.*2310

*It goes to 0.*2318

*The limit as x approaches 0 from below of e¹/x is actually equal to 0.*2323

*You get two different limits approaching a number one from above and one from below.*2333

*Again, it is all based on just asking yourself what happens to the function, or in this case, pieces of the function.*2339

*You use that piece to address the big function, when x does something.*2346

*Use your intuition, trust your intuition.*2351

*This is completely intuitive.*2354

*I know the x is negative.*2358

*Therefore, I know that 1/x is going to be a negative number.*2360

*If it gets closer and closer to 0, it is going to go off to infinity but it is going to go off to negative infinity.*2363

*e ⁻infinity is the same as 1/ e ⁺infinity.*2371

*As this denominator gets big, the function itself, it goes to 0.*2374

*Let us take a look and see what is this.*2382

*In this case, the limit as x approaches 0 of e¹/x does not exist.*2387

*It does not exist because you ended up with two different limits.*2397

*One is infinite, one is 0.*2400

*Again, we said, when you are approaching a number,*2402

*the left hand limit and the right hand limit have to be the same, in order for us to say that the limit exists.*2405

*The limit is this.*2410

*Our left hand limit exists at 0, the right hand limit hand limit exists in the sense that it goes off to infinity.*2415

*It is not a real number but they exist, it goes off to infinity.*2421

*But the limit itself does not exist.*2424

*Let me repeat, I know you are sick to death of me repeating this, I know.*2431

*You see that we are simply asking ourselves, *2437

*what happens to f(x) as x either approaches a number or as x approaches infinity.*2455

*That is all we are asking.*2472

*The reason I keep repeating myself, I apologize, is because sometimes*2473

*when you are just faced with some function, it is a little intimidating symbolically.*2477

*You sort of all of a sudden get a little intimidated and discombobulated, just what is it asking.*2481

*It is saying, as x does this, what does f(x) do?*2488

*Just remember that is all this symbolism is.*2493

*As x does this, what does f(x) do?*2496

*As long as you contain that, you can calm down and address the issue as you need to.*2499

*Essentially, it is simple.*2505

*The hardest part in taking limits is going to be the manipulation part.*2507

*How do I manipulate the function, in order to actually be able to take the limit and get something that makes sense.*2510

*It is always going to be the hardest part of calculus.*2517

*It is going to be the manipulation and the algebra, not the calculus itself.*2519

*In any case, that is that.*2523

*I will write this out.*2536

*If our first attempt is nonsense, when we take the limit nonsense, *2538

*then we manipulate and try again until we get f(x) actually approaches an actual number,*2549

*or we get f(x) going to positive or negative infinity.*2576

*If it does not work the second time, when you take the limit after you manipulate it, try another manipulation.*2580

*Try it again, you keep trying until one of these two things happens.*2584

*Either you end up with an actual number, when you take the limit, or you end up with positive or negative infinity.*2588

*That is when you can stop.*2592

*Let us try, this time, the limits as x goes to positive or negative infinity of e¹/x.*2598

*Now it is not x approaches 0, but it is x approaches positive or negative infinity of e¹/x.*2609

*Let us see what happens here.*2615

*As x approaches positive infinity, let us go ahead and write our function again here.*2623

*f(x) = e¹/x, actually let me write the entire limit.*2638

*We wanted to know what the limit as x approaches positive or negative infinity was, of e¹/x.*2645

*As x approaches positive infinity, 1/x goes to 0.*2651

*e¹/x, e⁰ approaches 1 from above, positive numbers.*2663

*As x approaches negative infinity, 1/x definitely approaches 0.*2675

*1/x approaches 0, with 1/x being negative values.*2700

*e¹/x is equivalent to 1/,*2710

*Let me make this more clear.*2727

*As x goes to negative infinity, this 1/x, it goes to 0 through negative values.*2728

*Because if x is a negative number then 1/x is a negative number, two negative values. *2735

*e⁻¹/x.*2749

*You know what, I do not want to do that.*2757

*We are still dealing with e¹/x.*2760

*e¹/x, except now this 1/x is our negative numbers.*2761

*That is going to be equal to 1/ e¹/x, where now these are positive.*2767

*e¹/x, we said that it approaches 1, this is going to approach 1/1.*2775

*It is going to be 1 from below.*2784

*The morale of all this is just keep track of all the details.*2788

*Just be really meticulous in what is it that you do and everything should work out.*2792

*Let us take a look at what this particular thing looks like.*2796

*This is our function.*2799

*Here our function is e¹/x.*2802

*As x gets really big positive, we said that the function itself is going to approach 1.*2806

*This dashed line is our horizontal asymptote, it is y = 1.*2813

*As x gets really big negative, 1/x is definitely going to be negative.*2820

*It is the same as 1/ e¹/x, it is going to also approach 1 from below.*2828

*But notice it never becomes negative because this is never a negative number.*2835

*It approaches it from below but still positive numbers, in the sense that.*2842

*1/x is negative, the e¹/x which is what this function is, it approaches 1 from below.*2847

*In this case, there is only one horizontal asymptote.*2854

*It is y = 1.*2857

*The lesson is basically this, there is no one way to evaluate every limit.*2868

*That is pretty much where it comes down to.*2888

*This is reasonably sophisticated mathematics.*2890

*As things become more sophisticated in mathematics, it is no longer going to be algorithmic.*2892

*There is a certain degree of algorithm, or if you do this and this, each problem is going to be different.*2898

*You have to bring all your resources to bare, whatever those resources are.*2903

*Whether they be graphs, tables, analytics, some trick that you learn when you are 10 years old, whatever it happens to be.*2907

*All of these things have to be brought to bare, as weapons against these particular problem.*2916

*There is no one way to evaluate every limit.*2921

*Do not think that you are supposed to just look at a problem and know how to do it.*2925

*The process of mathematics is not looking at it and just automatically knowing what to do.*2928

*If you fall into that trap, then when you are faced with something that is slightly different than what you used to, *2936

*you are not going to able to handle it.*2941

*Always try to keep a reasonable degree of objectivity.*2943

*Keep an open mind and do not worry that all of a sudden you look at something, *2945

*you do not know how to do it immediately.*2949

*Knowing how to do something immediately is not a measure of intelligence.*2952

*Intelligence has to do with being able to look at a situation and work the situation out.*2958

*This is sophisticated mathematics, it is not 1 or 2 steps.*2964

*There are going to be some problems in calculus over the next year that are multiple steps.*2967

*You are not going to even know where it is that you are going.*2972

*You are going to have to trust each step you take is a reasonable step.*2974

*You are going to have to hope that it is taking you somewhere.*2978

*That is what it is about, it is about getting to the answer in a reasonable logical way.*2980

*Even if you do not know exactly the path that you are following.*2985

*It could be a very circuitous path, in any case.*2988

*There is no one way to evaluate every limit.*2992

*Use every resource at your disposal.*2996

*Thank you so much for joining us here at www.educator.com.*2998

*We will see you next time, bye.*3000

1 answer

Last reply by: Professor Hovasapian

Fri Mar 25, 2016 11:09 PM

Post by Acme Wang on March 8 at 07:10:16 AM

Hi Professor Hovasapian,

I wanna ask some questions in Example III. The limit as x approaches positive infinity doesn't equal the limit as x approaches negative infinity, so can I say the limit for the equation does not exist? Kind of mixed up with the left-handed and right-handed limit.

Also, when x approaches infinity, does that indicate I must consider two circumstances (x approaches positive infinity and negative infinity)? Even when I take my AP exam?

Besides, in example III when x approaches positive infinity, you then wrote x>0? Why not x>1? Does xà+? means x>0?

Sincerely,

Acme