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 the Limit of a Function

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: Explain in Words What the Following Symbols Mean
- Example II: Find the Following Limit
- Example III: Use the Graph to Find the Following Limits
- Example IV: Use the Graph to Find the Following Limits
- Example V: Sketch the Graph of a Function that Satisfies the Following Properties
- Example VI: Find the Following Limit
- Example VII: Find the Following Limit

- Intro 0:00
- Example I: Explain in Words What the Following Symbols Mean 0:10
- Example II: Find the Following Limit 5:21
- Example III: Use the Graph to Find the Following Limits 7:35
- Example IV: Use the Graph to Find the Following Limits 11:48
- Example V: Sketch the Graph of a Function that Satisfies the Following Properties 15:25
- Example VI: Find the Following Limit 18:44
- Example VII: Find the Following Limit 20:06

### AP Calculus AB Online Prep Course

### Transcription: Example Problems for the Limit of a Function

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

*Today, we are going to do some example problems for the limit of a function.*0004

*Let us jump right on in.*0009

*The first one says, explain in words what the following symbols mean.*0012

*The limit as x approaches 6 of f(x) = 14, what does this mean?*0016

*It means the following.*0023

*I think I will work in red here.*0026

*This one says, as x gets arbitrarily close to 1,*0028

*I do not like arbitrarily close, let us just say closer and closer indefinitely.*0037

*As x gets closer and closer to 6 from below and above,*0042

*because we do not have that positive or negative superscript on the number, f(x) gets closer and closer to 14.*0058

*That is all this symbol says.*0076

*As x gets arbitrarily close to 6 from below and from above, f(x) is getting close to 14 from below and from above.*0078

*How about this one, the limit as x approaches 4 from below of f(x) = infinity?*0092

*This one says, as x gets close to 4 from below, from the left, it is going to be interchangeable.*0098

*Sometimes, we will say from below.*0114

*Sometimes, we will say from the left.*0115

*Sometimes, we will just say left hand limit, things like that.*0117

*They are all synonymous.*0119

*As x gets close to 4 from the left, f(x) goes off to positive infinity.*0121

*This one, the limit as x approaches 4 from the right is equal to -3.*0138

*This one says, as x approaches 4 from the right, this time I will go ahead and write from above,*0145

*all of these things is synonymous, f(x) approaches the number -3.*0169

*It gets closer and closer to, approaches, all of these words are going to be used.*0185

*Given what you have just written, for the first limit, this one, can f(x) equal 5?*0191

*The limit as x approaches 6 of f(x) is 14, can f(x) = 5?*0200

*Absolutely, yes, it is not a problem at all because we know that the limit of a function,*0207

*as it gets close to a number is independent of the value of the function at the number.*0212

*Yes, not a problem of all.*0220

*The limit as x approaches 6 of f(x).*0228

*It is starting to get a little crazy now with my writing.*0237

*Is independent of what happens when x = 6.*0242

*Draw all possible graph for the second limit above.*0261

*The second limit above that is that one right there.*0268

*The limit as x approaches 4 from below, this is not as x approaches -4.*0274

*As x approaches 4 from below, from the left of f(x) goes off to infinity.*0281

*We have something like this.*0286

*Let us say this is 4, we are going to approach 4 from below which means we are going to approach it this way.*0288

*We are going to take x value getting closer and closer, and then, it is going to go off to positive infinity.*0296

*This positive infinity, this is an asymptote.*0301

*This might be a graph right there.*0303

*As it gets closer and closer to 4, the function blows up to infinity.*0308

*That is it, very straightforward, just go with your intuition.*0313

*Clarify the following limit if it exists.*0324

*The limit as x approaches 0 of sin x/ x.*0327

*As of right now, you have two intuitive tools that you can use, in order to find the limit.*0331

*You have the graph of the function and you can make a table of values.*0335

*Taking values closer and closer to whatever number that you are approaching.*0340

*You can use both, and both are very powerful tools.*0344

*Clearly, the tabular gives you a better idea of what is happening because it gives you specific numbers.*0347

*But the graph is perfect because it gives you a nice first approximation for what is happening.*0352

*In this particular case, as x approaches 0, notice we are not specifying left hand or right hand limit.*0357

*We have to do both.*0363

*We have to approach 0 from the left from below and we have to approach it from the right from above.*0365

*In each case, when we look at the graph, the function looks like it is approaching 1.*0373

*And then from the right, the function is climbing and climbing, it looks like it approaches 1.*0381

*Let us go ahead and confirm that with our table of values.*0386

*From here to here, this is x approaching 0 from below -0.5, -0.25, 0.1, -0.01, -0.001.*0389

*We see it is going from 0.95, 0.9999, definitely it looks like it is getting close to 1.*0402

*Here x is approaching 0 from above 0.5, 0.25, 0.1, 0.01, 0.001.*0413

*We are getting closer and closer and closer to 0.*0423

*We see that the value of the function itself is going 0.95, 0.98, 0.998, 0.99999999999.*0426

*Yes, we can conclude that this limit is equal to 1.*0437

*You use your graph, you use your table of values.*0446

*In a subsequent lesson, we are going to learn how to do this analytically.*0449

*Example 3, use the graph below of f(x) to find the following limits, if they exist.*0457

*If they do not exist, give the reason why.*0463

*This is our function right here, we see it down below.*0465

*We have solid dot here.*0469

*Any place where on these graphs where you actually do not see an open dot or open, this is an open circle here.*0473

*It is going to be an open circle here.*0480

*Let me actually work in blue.*0482

*We see that at 3, the value of the function has actually defined what is that solid dot.*0485

*It is over here at 2.5.*0490

*The limit as x approaches 1.5 of f(x), 1.5 is here.*0494

*It does not specify, we have to approach it this way and approach it this way.*0502

*What happens to the function?*0506

*The function looks like it is getting close to 4.*0508

*From here, it looks like f is getting close to 4.*0512

*This limit is equal to 4, nice and simple, very straightforward.*0516

*The e to the left and right hand limits equal each other.*0527

*The limit exist and the limit = 4.*0529

*The limit as x approaches 3.*0532

*The limit as x approaches 3 from here, let us actually do c and d first because here it says 3 from below and 3 from above.*0536

*These two together are the same as what b is.*0544

*The limit as x approaches 3 from below, following this way, the y values looks like it is going to be right about there.*0548

*That looks like it is going to be about 3. 15, something like that.*0561

*Let us say this approximately equals like 3.2, and then, the limit as x approaches 3 from above.*0571

*When we come this way, as we get close to 3 from this way, the function is getting close to 1.*0580

*This one is equal to 1.*0588

*The left hand limit 3.20, right hand limit 1.*0591

*Therefore, this limit does not exist.*0595

*The left hand limit exists, the right hand limit exists, but because these are not equal, the limit itself does not exist at 3.*0598

*The limit as x approaches -2 from above.*0608

*Notice, this negative in front, that is the number.*0612

*The positive, that is from above.*0615

*We are approaching -2 from above.*0618

*What is the function doing, it looks like it is getting close to that number right there.*0622

*It looks like that number is about, let us say 2.1.*0626

*What is f(3)?*0633

*The value of f(3) is 2.5.*0634

*Notice the left hand limit is about 3.2.*0641

*The right hand limit, the limit from above is 1.*0652

*The value of the function at 3 is 2.5.*0655

*These are all different, they are independent.*0658

*You can have a left hand limit, you can have a right hand limit.*0664

*You are going to have the value of the function at that point.*0666

*They do not have to be equal.*0670

*They are incompletely independent.*0671

*If the two limits are equal, we say the limit exists.*0673

*If the two limits are equal and it happens that equal the value of the function, then we say that the function is continuous.*0677

*That is a stronger property.*0685

*Here, clearly, the function is discontinuous.*0688

*Because you see that I’m drawing my curve, I have to lift my pencil and that continue on.*0690

*This is a discontinuity.*0694

*If all three are equal which we will get to, we actually call that continuous.*0695

*That is a very special property, when all three are equal.*0700

*Let us go ahead and try another one.*0707

*Use the graph below of f(x) to find the following limit, if they exist.*0712

*Same thing that we just did, if they do not exist, give the reason why.*0715

*The limit as x approaches a -1.*0719

*-1 is here, x approaches -1, it does not specify.*0726

*We have to do a left hand limit and we have to do a right hand limit.*0731

*The left hand limit, as x approaches, we see that the function approaches this value right there, whatever that happens to be.*0735

*The right hand limit as we approach -1 from the right, the function is this.*0751

*The limit is here.*0757

*Clearly, this number and this number are not the same.*0759

*The limit as x approaches negative of f(1), it does not exist.*0763

*What is the value of f(-1), it is this one right here.*0769

*It is actually equal to 1.*0775

*What is the limit as x approaches 1 from below?*0778

*1 from below, the limit is 1.*0783

*The limit as x approaches 1 from above.*0792

*From above, the function is approaching -1.*0795

*A left hand limit exists, a right hand limit exists.*0803

*The limit itself does not exist.*0806

*What is the value of f(1)?*0809

*It turns out that the left hand limit corresponds with the value of the function at the point.*0814

*Notice, this is equal to that but it is not equal to this.*0824

*This is not continuous.*0827

*You can see from the graph that it is not continuous.*0828

*If you did not have a graph, if you just had this and this, you could say that is not continuous.*0830

*You do not need the graph, the limits will tell you whether something is continuous or not.*0838

*The limit as x approaches 3 of f(x).*0844

*3 is over here, we are going to approach it from below.*0846

*As we approach it from below, it looks like the graph is approaching -1.*0852

*As we approach it from above, it approach the value 3 from above, it is x does this.*0858

*The value of the function itself approaches -1.*0864

*Here, the left hand limit which is -1, right hand limit -1.*0868

*They equal, the limit = -1.*0875

*In this case, the value of the function at 3, in other words, f(3) also happens to equal -1.*0877

*When all three are equal, this is a continuous function.*0885

*As you can see, this is a nice continuous function.*0888

*It is perfectly smooth, in other words.*0892

*When we say it is continuous, we mean smooth.*0894

*There is no break, I do not have to lift my pencil to continue the graph.*0897

*This is a discontinuous function, there is a discontinuity at -1 and there is a discontinuity at +1.*0902

*There are left hand and right hand limits, they do not equal each other.*0910

*A left hand and right hand limit that do not equal each other.*0914

*At 3, it is continuous.*0917

*The left hand limit, the right hand limit, and the value of the function equal each other.*0919

*Sketch the graph of the function that satisfies the following properties.*0926

*Very simple, you just have to make a graph that satisfies these properties.*0931

*The limit as x approaches 0 from below is negative infinity.*0937

*The limit as x approaches 0 from above is positive infinity.*0944

*I know the graphs goes, let me do the graph in red actually.*0947

*Negative infinity, this one, as I approach 0 from below, my graph goes to negative infinity.*0951

*As I approach 0 from above, it goes to positive infinity.*0957

*The limit as x goes to negative infinity, as x gets really big in the negative direction is 0.*0964

*I know that it gets close to here, maybe not.*0970

*The limit as x approaches 2 of f(x) is equal to 3.*0978

*The limit as x approaches positive infinity is equal to 1.*0990

*Let us deal with this one, as x approaches positive infinity.*0997

*Let us stay over here on the right side of the graph, +1.*1001

*I’m going to go ahead and draw an asymptote at 1.*1005

*It tells me that as x goes to positive infinity, f(x) gets close to 1.*1011

*I’m going to go ahead and draw that.*1016

*Here as x approaches negative infinity, f(x) goes to 0.*1023

*I know I'm here, this is going to go probably, it is a possibility.*1036

*It is not the only possibility but this is the possibility.*1041

*f(2) is undefined.*1044

*As x approaches 2, it equal 3.*1047

*Let us say this is 2, let us say this is 3.*1049

*I have to actually connect it.*1058

*It is undefined there, there is going to be a hole there.*1059

*But the limit as x approaches 2, meaning from the left and from the right, the limit is the same.*1063

*The function actually touches here and goes down that way.*1070

*There we go, that is a much better function.*1075

*That satisfies all of the properties.*1078

*f(2) is undefined.*1080

*As x approaches 0 from below, the graph goes to negative infinity.*1084

*As x approaches 0 from above, the graph goes to positive infinity.*1089

*As x goes to negative infinity, the graph goes to 0.*1093

*Yes, the y value goes to 0.*1096

*As x approaches 2 from below, from above the graph, it approaches the number 3.*1099

*That takes care of that.*1106

*As x goes to infinity, f(x) approaches the value 1.*1108

*Yes, that is the horizontal asymptote and f(2) is undefined.*1112

*This is a graph, not the only graph, there might be others but that is a good graph.*1116

*Find the following limit if it exists.*1127

*The limit as x approaches 0 of this function.*1128

*Here, we have the graph of the function.*1133

*We are going to use our graph to let us know what is happening as x approaches 0.*1135

*X approaches 0, 0 is a number.*1139

*It does not specify whether it is left or right hand limit, we have to do both.*1142

*We have to approach 0 from below, 0 from above.*1146

*As we approach 0 from below, the limit of f(x) as x approaches 0 from below.*1152

*As we approach it from below, the function, the y value goes to positive infinity.*1160

*The limit as x approaches 0 from above of this f(x).*1169

*As we go this way, the function goes to negative infinity.*1173

*The limit does not exist.*1180

*That is it, nice and simple.*1183

*Just use the graph to tell you what is going on.*1185

*If you need a table of values, you can use a table of values.*1187

*But here, it is clear, you can see what the graph is doing.*1190

*They are blowing up to infinity but in opposite directions.*1194

*If this 1 to infinity from the left and from the right, we can say the limit = positive infinity.*1197

*Determine the following limit.*1211

*The limit as x approaches 5 of sin x × ln x.*1212

*Here we avail ourselves of both the graph and the table of values.*1219

*From the graph, we are approaching 5.*1231

*This is the number we are approaching.*1233

*We are approaching it from this point and we are approaching if from the right.*1234

*From below, from above.*1238

*Left hand limit, right hand limit.*1240

*The left hand limit, as it approaches this number, we see that it gets close to some value.*1244

*As we approach x, we approach 5 from the right.*1250

*We see that the y value approaches the same value.*1253

*Let us see what it is at, here, x approaches 5 from below.*1257

*Here this is x approaching 5 from above.*1264

*Let us see what the numbers do.*1268

*4.9, 4.99, 4.999, 4.9999, getting really close to 5.*1270

*We notice that we are approaching -1.5433.*1276

*Here, same thing, we go to 5.1, 5.01, 5.001, 5.0001, getting closer and closer.*1284

*We see that we are approaching the same number 1.5433, -1.5433.*1298

*I can conclude, because the left hand limit and the right hand limit are the same,*1307

*they look like they are the same number.*1313

*Clearly, they are the same in the graph.*1315

*The limit as x approaches 5 of the sin(x) × natlog(x).*1317

*I can say that it equal -1.543.*1327

*It is good to 3 decimal places.*1332

*That is it, very intuitive notion actually.*1336

*Yet, the entire modern world is based on this notion.*1343

*The idea of the limit, the idea of the derivative.*1346

*It is quite extraordinary.*1348

*Let us go ahead, given f(x), let us round out what we have done here today.*1353

*Given f(x), use your calculator or a graphing utility*1360

*or graphing software to make a graph of the function that you are dealing with.*1379

*Make a table of values and speculate about whether the limit converges, whether the function converges to a specific number.*1395

*Use your graph, use your table of values and make some good choices.*1434

*In the next lesson, we learn how to evaluate limits analytically.*1440

*In other words, doing something mathematical.*1467

*It is going to turn out that the math is really simple.*1474

*Thank you for joining us here at www.educator.com.*1480

*We will see you next time, bye.*1482

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