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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### Example Problems for Limits at 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
- Example I: Calculating Limits as x Goes to Infinity
- Example II: Calculating Limits as x Goes to Infinity
- Example III: Calculating Limits as x Goes to Infinity
- Example IV: Calculating Limits as x Goes to Infinity
- Example V: Calculating Limits as x Goes to Infinity
- Example VI: Calculating Limits as x Goes to Infinity

- Intro 0:00
- Example I: Calculating Limits as x Goes to Infinity 0:14
- Example II: Calculating Limits as x Goes to Infinity 3:27
- Example III: Calculating Limits as x Goes to Infinity 8:11
- Example IV: Calculating Limits as x Goes to Infinity 14:20
- Example V: Calculating Limits as x Goes to Infinity 20:07
- Example VI: Calculating Limits as x Goes to Infinity 23:36

### AP Calculus AB Online Prep Course

### Transcription: Example Problems for Limits at Infinity

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

*Today, I thought we would do some more example problems for limits at infinity*0004

*or as x goes to positive or negative infinity.*0009

*Let us jump right on in.*0011

*Evaluate the following limit x² + 6x - 12/ x³ + x² + x + 4, as the limit as x goes to negative infinity.*0015

*Here we have a rational function.*0029

*We know how to deal with rational functions.*0031

*We pretty much just divide the top and bottom, by the highest power of x in the denominator.*0035

*The first thing you do is basically plug in.*0048

*You evaluate the limit to see, before you actually have to manipulate.*0050

*In this case, when we put in negative infinity into here and here, what we are going to end up with is,*0054

*I will say plugging in, we get infinity/ negative infinity.*0062

*x², this term is going to dominate, this term is going to dominate.*0083

*Negative infinity², negative number² is positive.*0089

*Negative number³ is going to be negative.*0092

*We are going to get something like this which does not make sense.*0094

*This is going to be the limit as x goes to negative infinity of x² + 6x - 12/ the greatest power in the denominator which is x³.*0100

*That is our manipulation x³ + x² + x + 4/ x³.*0112

*This gives us the limit as x approaches negative infinity of, here we have 1/ x + 6/ x² – 12/ x³ / 1 + 1/ x + 1/ x² + 4/ x³.*0121

*Now when we take the limit as x goes to negative infinity, this is 0, this is 0, this is 0.*0152

*All of these go to 0 and you are left with 0/1 which is an actual finite number 0.*0159

*Our limit is 0.*0166

*Again, it is always nice to confirm this with a graph.*0169

*This is our function, down here a little table of values.*0177

*Basically, what this part of the table of values does is it confirms the fact that we start negative,*0181

*the function crosses the 0.*0186

*And then actually comes up and gets closer and closer to 0 which is the limit.*0191

*As x gets really big, the function gets close to 0 which is what we just calculated analytically.*0199

*Evaluate the following limit.*0209

*The limit as x goes to infinity of x + 4/ 8x³ + 1.*0211

*A couple of things to notice.*0218

*Again, instead of just launching right in this, you want to stop and ask yourself some questions.*0219

*Here the radical is the denominator.*0223

*It is a rational function.*0226

*We have to think about these things.*0227

*8x³ + 1, it is under the radical sign.*0229

*It, itself has to be greater than 0.*0233

*Let us see what the conditions are here.*0236

*Here the 8x³ + 1 is in the denominator.*0241

*We know that that value cannot be 0.*0251

*It cannot equal 0.*0257

*Since we cannot have the square root of a negative number, 8x³ + 1 has to be greater than 0.*0266

*It could be greater than or equal to 0.*0275

*But then, if it were equal to 0 then you have a 0 in the denominator.*0277

*That is why we have only the relation greater than.*0279

*I will write not greater than or equal to, which normally we could.*0283

*Let us go ahead and work this out first.*0288

*8x³ + 1 is greater than 0.*0290

*We have 8x³ greater than -1.*0295

*We have x³ greater than -1.*0298

*That is greater than -1/8, which implies that x itself has to be greater than -1/2.*0307

*That is our domain.*0317

*We said not that relation, greater than -1/2.*0320

*Here we do not have to worry about going to negative infinity.*0324

*All we have to worry about here is going to positive infinity.*0328

*Because x cannot be, x cannot go to negative infinity.*0331

*We do not have to worry about x going to negative infinity.*0339

*We saved ourselves a little bit of work.*0352

*Our x + 4/ 8x³ + 1, it is a rational function.*0356

*We want to divide it by the greatest power of x in the denominator.*0368

*This is going to be x + 4, x³/ √x³.*0379

*This is going to be x + 4/ x³/2.*0396

*The √x³ is x³/2 / 8x³ + 1/ x³.*0405

*That is going to equal, this is going to be 1/ x ^½.*0418

*This is going to be + 4/ x³/2.*0424

*This is going to be √8 + 1/ x³.*0429

*Now we go ahead and take the limit.*0440

*The limit as x goes to infinity of 1/ x ^½ + 4/ x³/2 / √8 + 1/ x³.*0445

*As x goes to infinity, this goes to 0, this goes to 0, this goes to 0.*0463

*We end up with 0/ √8 which is a finite number.*0468

*Our limit is 0.*0472

*The graph that we get is this.*0477

*We see as x gets really big, the function gets closer and closer and closer to 0.*0480

*There we go, confirmed it graphically.*0487

*Evaluate the following limit.*0492

*The limit of √9x ⁺10 – x³/ x + 5 + 100.*0494

*The best way to handle this is, let us try this.*0503

*Again, my way is not the right way.*0511

*It is just one way, you might come up with, 5 different people might come up with 5 different ways of doing this limit.*0513

*That is totally fine, that is the beauty of this.*0518

*Let us do the following.*0531

*Let us take this 9x ⁺10 – x³.*0533

*As x goes to infinity, here only 9x ⁺10 term is going to dominate.*0541

*I’m just going to deal with that term.*0557

*The same thing for the denominator.*0559

*For the denominator, the x⁵ is the one that I'm going to take.*0560

*Basically, what the limit that I'm going to take is the limit as x approaches infinity of √9x ⁺10/ x⁵.*0569

*Essentially, I just said that this does not matter and this does not matter.*0582

*The limit is going to be essentially the same.*0585

*I just deal with those.*0587

*This is going to equal to the limits as x approaches infinity, √9x ⁺10 = absolute value of 3x⁵.*0591

*Remember, the square root of something is the absolute value of something/ x⁵.*0608

*For x greater than 0, in other words x going to positive infinity.*0618

*The absolute value of 3x⁵/ x⁵, when x is greater than 0, this is just 3x⁵/ x⁵ = 3.*0625

*The limit as x approaches positive infinity of 3 = 3.*0640

*Now for x less than 0, this 3x⁵ absolute value/ x⁵, it is actually going to be -3x⁵/ x⁵ = -3.*0648

*The limit as x goes to negative infinity of -3 = -3.*0668

*Here you are going to end up with two different asymptotes, 3 and -3.*0678

*Now notice the difference between the following.*0687

*We finished the problem but we are going to talk about something.*0693

*Notice the difference between following.*0700

*When I take √x ⁺10, I get the absolute value of x⁵.*0708

*The absolute value of x⁵ is either going to be x⁵ because when x is greater than 0 or it is going to be,*0717

*I actually did it in reverse, that is why I got a little confused here.*0740

*It is going to be –x⁵ and that is when x is less than 0.*0742

*When x is greater than 0, it is going to be x⁵.*0751

*The reason is because this is an odd power.*0755

*x⁵ itself, depending on whether x is positive or negative, this inside is going to be a positive or negative number.*0768

*If what is in here is positive, then it is going to go one way.*0780

*If what is in here is negative, it is going to go the other way.*0785

*However, if I take something like x⁸, this is going to give me,*0788

*we said that the square root of a thing is going to be absolute value of x⁴.*0798

*Here it does not matter.*0804

*If x is positive or negative, it is an even power.*0806

*An even power is always going to be positive number.*0812

*Therefore, this is just going to be x⁴ because it is an even power.*0816

*Be careful of that, you have to watch the powers.*0823

*When you pass from the square root of something, we said the square root of something is the absolute of something.*0825

*But the power itself is going to make a difference on whether you separate or whether you do not.*0831

*Let us take a look at the graph of the function that we just did.*0838

*We said that as x goes to positive infinity, the function approaches 3.*0842

*As x goes to negative infinity, the function approaches -3.*0849

*I did not draw out the horizontal asymptotes here.*0854

*I just want you to see that, but it is essentially what we did.*0855

*Evaluate the following limit, the limit as x approaches positive infinity of 16x² + x.*0863

*You might be tempted to do something like this.*0875

*Let me write this down.*0877

*We are tempted to say as x goes to positive infinity, this term is going to dominate, which is true.*0879

*We are tempted to say that we can treat this as √16x² which is equal to 4x.*0898

*We can just say that 4x - 4x is equal to 0.*0909

*The limit is x approaches this is just 0.*0913

*That is not the case.*0917

*The problem is the x term may contribute to such a point that, what you end up with is infinity – infinity.*0919

*This will go to infinity, this will go to infinity.*0944

*But infinity – infinity, we are not exactly sure about the rates at which this goes to infinity and this goes to negative infinity.*0946

*Because we are not sure about how fast that happens,*0955

*we do not know if it goes to 0 or if it goes to infinity, or if it goes to some other number in between.*0957

*This is an indeterminate form, infinity – infinity.*0964

*We have to handle it differently.*0967

*Let us deal with the function itself, before we actually take the limit.*0970

*Here when we put the infinity in, we get infinity - infinity which does not make sense.*0974

*We have to manipulate it.*0979

*We have 16x² + x under the radical, -4x.*0982

*I’m going to go ahead and rationalize this out.*0989

*I’m going to multiply by its conjugate.*0991

*16x² + x, this is going to be + 4x/ 16x² + x + 4x.*0994

*When I multiply this out, I end up with 16x² + x.*1007

*This is going to be -16x²/ √16x² + x/ +4x.*1013

*Those go away, leaving me with just x/ 16x² + x + 4x.*1027

*We have a rational function, even though we have a square root in the denominator,*1040

*let us go ahead and divide by the largest power in the denominator which is what we always do in the denominator.*1043

*The largest power in the denominator is essentially going to be the √x².*1060

*It is going to be x, but it is going to be √x².*1065

*What we have is the following.*1069

*We are going to have x/x, that is the numerator.*1072

*We are going to have 16x² + x, all under the radical, + 4x all under x, which is going to equal 1/ √16x² + 4x/ x².*1075

*This one, I’m going to treat x as √x².*1115

*I’m going to get 16x² + 4x/ x², + 4x/ x, I’m going to leave this as x.*1121

*This x for these two, because under the radical I’m just going to treat it as √x².*1132

*That ends up equaling 1/ √16, this is x.*1141

*+ 1/ x and this is + 4.*1155

*There we go, now we can take the limit.*1160

*The limit as x approaches positive infinity of 1/ √16 + 1/ x + 4.*1164

*As x goes to infinity, the 1/x goes to 0.*1175

*We are left with 1/ 4 + 4.*1181

*√16 is 4, you will get 1/8.*1184

*Sure enough, that is what it looks like.*1192

*This is our asymptote, this is y = 1/8.*1194

*This is our origin, as x goes to positive infinity, the function itself gets closer and closer to 1/8.*1198

*That is the limit.*1206

*Evaluate the following limit as x goes to infinity, 1 – 5e ⁺x/ 1 + 3e ⁺x.*1210

*When we put x in, we are going to end up with -infinity/ infinity which is in indeterminate form.*1220

*We have to do something with it.*1231

*1 – 5e ⁺x/ 1 + 3e ⁺x, we can do the same thing that we did with rational functions.*1235

*This is going to be the same as, I’m going to divide everything by e ⁺x.*1244

*1 - 5e ⁺x, the top and the bottom, I mean, / 1 + 3e ⁺x/ e ⁺x.*1249

*What I end up with is 1/ e ⁺x - 5/ 1/ e ⁺x + 3.*1258

*I’m going to take the limit of that.*1272

*The limit as x goes to, I’m going to do positive infinity first.*1274

*1/ e ⁺x – 5, put the 1/ e ⁺x + 3.*1283

*As x goes to infinity, e ⁺x goes to infinity that means this thing goes to 0, this thing goes to 0.*1291

*I'm left with -5/3.*1297

*As x goes to positive infinity, my function actually goes to -5/3.*1299

*I have a horizontal asymptote at 5/3, -5/3.*1306

*Now for x going to negative infinity, I have the following.*1311

*The limit as x goes to negative infinity of 1 - 5e ⁺x/ 1 + 3e ⁺x.*1318

*x is a negative number, it is negative infinity.*1332

*e ⁺negative number is 1/ e ⁺positive number.*1336

*This is actually equivalent to the limit as x goes to positive infinity of 1 - 5/ e ⁺x.*1339

*It is e ⁻x is the same as e ⁻x is 1/ e ⁺x.*1357

*Because we are going to negative infinity, x is negative number.*1366

*Because it is a negative number, I can just drop it into the denominator and make it a positive number.*1372

*1 - 5 and then 1 + 3/ e ⁺x.*1377

*As x goes to infinity, this goes to 0, you are left with 1.*1383

*Sure enough, there you go.*1392

*As x goes to negative infinity, we approach y = 1.*1394

*As x goes to positive infinity, our function approaches y = -5/3.*1401

*That is it, just nice manipulation.*1410

*Let us see what we got.*1417

*Now the whole idea of a reachable finite numerical limit is as x gets closer and closer to a certain number or as x goes to infinity,*1418

*but f(x) gets closer and closer to a certain number like we just saw -5/3 or 1.*1427

*This latter number is the limit.*1434

*The question here is how big would x have to be, in order for the function f(x) = e ⁻x/25 + 2*1439

*to be less than a distance of 0.001 away from its limit?*1449

*Closer and closer, closer and closer means we can take it as close as we want.*1455

*In this case, the tolerance that I'm looking for is 0.001 away from its limit.*1459

*The first thing we want to do, what is the limit?*1467

*What is the limit as x goes to infinity of e ⁻x/ 25 + 2.*1474

*Let us just deal with positive infinity here.*1489

*This is the same as e ⁻x/25.*1507

*This is the same as the limit as x goes to positive infinity of 1/ e ⁺x /25 + 2.*1512

*As x goes to infinity, e ⁺x/25 goes to infinity.*1532

*This goes to 0.*1537

*The limit is actually 2.*1542

*The limit of this function as x goes to infinity is equal to 2.*1544

*I probably going to need more room.*1557

*Let me go ahead and go and work in red.*1559

*Now e ⁻x/25 is always greater than 0.*1566

*e ^- x/ 25 + 2 is always going to be greater than 2.*1576

*f(x) which is equal to e ^- x/25 + 2, we said that the limit of this function as x goes to infinity is 2.*1592

*But we said that the function is always greater than 2 which means that*1602

*the function is actually approaching 2 from above.*1606

*It actually looks like this, this is our graph and this is our asymptote at 2.*1610

*The function is doing this.*1619

*The limit is 2, that is this dash line right here.*1622

*We know that the function itself, because this is always greater than 0, the function itself e ⁻x/25 + 2 is always going to be greater than 2.*1627

*It is always going to be above it.*1635

*It is above it, it is getting closer to it from above.*1637

*That is what is happening here physically, getting closer to the 2.*1640

*That is happening from above.*1645

*We want to make this distance, that distance right there.*1648

*We will call it d, we want that distance to be less than 0.001.*1651

*Our question is asking how far out do we have to go?*1659

*What x value passed which x value will this distance?*1663

*This distance between the function and limit be less than 0.001, that is what this is asking.*1668

*Again, we said we will call that d.*1680

*D is equal to the function itself - the limit.*1685

*Here was the limit, here was the function, this is the distance right there.*1694

*We want that distance, that distance is f(x) – l.*1698

*Here is our origin, this is 0,0.*1704

*This number - this number gives me the distance between them.*1708

*It is f(x) – l.*1712

*We know what f(x) is, that is just e ⁻x/25 + 2.*1714

*We know what l is, it is -2.*1719

*These go away, we want this distance which is e ⁻x/25, we want it to be less than 0.001.*1724

*Now we can solve this equation for x.*1737

*I’m going to go ahead and take the natlog of both sides.*1744

*I have -x/25 is less than the natlog of 0.001.*1750

*I’m going to make this a little more clear here, 0.001.*1765

*-x is less than 25 × the natlog of 0.001, that means x is greater than -25 × the natlog of 0.001.*1771

*Whenever I do, the natlog of 0.001 is going to be a negative number.*1790

*Negative × a negative, when I put this in the calculator, I get x has to be greater than 172.069.*1795

*The limit was this number.*1808

*The whole idea of the limit is we want to get closer and closer and closer.*1810

*In this particular case, we specified what we meant by closer and closer.*1813

*I want it closer than 0.001, the function to be less than not far away from the limit.*1817

*I knew that the function was approaching it from above.*1823

*The distance between the function and limit, that is what I want it to be, less than 0.01.*1827

*The distance between the function and limit is the function - the limit.*1832

*This distance, that distance right there.*1836

*I set it and I solve for x.*1839

*As long as x is bigger than 172.69, f(x) - l is going to be less than 0.001.*1841

*In other words, the function is going to be less than 0.001 units away from its limit.*1853

*What if f actually approached it from below?*1865

*What if we have something like this?*1868

*Let us say again, this was our limit.*1874

*This time let us say that the function came from below.*1877

*Now this is f(x) and this is the origin.*1880

*The limit is above the function.*1884

*The distance that we are interested in is this distance.*1889

*The distance between the function and limit.*1891

*Here the distance is going to equal the length - the function.*1894

*The length is a bigger number.*1902

*We want it to be positive.*1905

*It is going to be l – f(x).*1907

*Now we combine f(x) - l and l - f(x) as the absolute value of f(x) – l.*1910

*This distance, and if we are coming from above, this distance,*1927

*they will be the same if we use the absolute value because distance is a positive number.*1930

*You cannot have a negative distance.*1936

*We just combine those two, when we give the definition of a limit by using the absolute value sign.*1938

*It is that absolute value sign that has confounded and intimidated the students for about 150 years now.*1944

*Our formal mathematical definition of the limit.*1955

*We are concerned with the formal mathematical definition of the limit.*1975

*I told you not to believe about that, I do not believe that these kind of definitions,*1977

*these precise definitions do not belong in this level.*1984

*This level is about intuition.*1986

*By intuition, we mean the idea of how close can you get.*1988

*We speak of closer and closer and closer.*1992

*There is a way of describing that symbolically.*1995

*What do we mean by closer and closer, that is what I'm going to describe here.*1997

*Again, I just want you to see it because some of your classes will deal with it,*2001

*some of your classes would not deal with it.*2004

*But I wanted you to see the idea and where it actually came from.*2006

*Our formal mathematical definition of, when we say something like the limit as x approaches infinity of f(x) = l.*2011

*When we say that some function as x goes to infinity,*2023

*that the function actually approaches a finite limit, this symbol, here is what it means mathematically.*2027

*The formal definition is for any choice of a number that we will symbolize with ε,*2035

*which is going to be always greater than 0, there is an x value somewhere on the real line.*2052

*Such that the absolute value of f(x) - the limit is going to be less than this choice of ε,*2065

*whenever x is greater then x sub 0.*2076

*In the previous example, we found our x sub 0, that was our 172.*2080

*We found an x sub 0.*2090

*Our ε in that problem, we chose 0.001 as our ε.*2093

*We wanted to make the difference between the function and the limit less than 0.001.*2100

*We found an x sub 0 of 172.69.*2104

*Why do I keep writing 6, that is strange.*2110

*Any x value that is bigger than 172.69, we will make the difference between the function and the limit less than 0.001.*2115

*The precise general mathematical definition is, for any choice of the number ε greater than 0,*2124

*there exists an x sub 0 such that whenever x is bigger than x sub 0,*2131

*the difference between f(x) and its limit is going to be less than ε.*2138

*You can see why this stuff is confusing and why is it that it actually does not belong at this level.*2142

*Again, for those of you that go on in mathematics and taking analysis course, math majors mostly, this is what you will do.*2147

*You will go back and you actually work with epsilons, deltas, and x sub 0.*2154

*You will prove why certain things are the way they are.*2159

*At this level, we just want to be able to accept that those proofs had been done.*2162

*We want to be able to use it to solve problems.*2166

*We want to learn how to compute.*2169

*We want to use it as a tool.*2171

*We do not want to justify it.*2173

*Later, you can justify it, as a math major.*2174

*Now we just want to be able to use it.*2177

*This idea of closer and closer and closer to a limit is absolutely fine.*2179

*It is that intuition, if you want to do it.*2185

*Thank you so much for joining us here at www.educator.com.*2188

*We will see you next time, bye.*2190

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