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For the function $f$ whose graph is given, determine the following limits.a. $\lim _{x \rightarrow 4} f(x)$b. $\lim _{x \rightarrow 2^{+}} f(x)$c. $\lim _{x \rightarrow 2^{-}} f(x)$d. $\lim _{x \rightarrow 2} f(x)$e. $\lim _{x \rightarrow-3^{+}} f(x)$f. $\lim _{x \rightarrow-3^{-}} f(x)$g. $\lim _{x \rightarrow-3} f(x)$h. $\lim _{x \rightarrow 0^{+}} f(x)$i. $\lim _{x \rightarrow 0^{-}} f(x)$j. $\lim _{x \rightarrow 0} f(x)$k. $\lim _{x \rightarrow \infty} f(x)$1. $\lim _{x \rightarrow-\infty} f(x)$

a) $$\lim _{x \rightarrow 4} f(x)=2$$b) $$\lim _{x \rightarrow 2^{+}} f(x)=-3$$c) $$\lim _{x \rightarrow 2^{-}} f(x)=1$$d) $$\lim _{x \rightarrow 2} f(x)$$ does not existe) $$\lim _{x \rightarrow-3^{+}} f(x)=\infty$$f) $$\lim _{x \rightarrow-3^{-}} f(x)=\infty$$g) $$\lim _{x \rightarrow-3} f(x)=\infty$$h) $$\lim _{x \rightarrow 0^{+}} f(x)=\infty$$i) $$\lim _{x \rightarrow 0^{-}} f(x)=-\infty$$j) $$\lim _{x \rightarrow 0} f(x)$$ does not existk) $$\lim _{x \rightarrow \infty} f(x)=0$$k) $$\lim _{x \rightarrow-\infty} f(x)=-1$$

01:21

Subhadeepta S.

Calculus 1 / AB

Chapter 2

Limits and Continuity

Section 6

Limits Involving Infinity; Asymptotes of Graphs

Limits

Derivatives

Harvey Mudd College

Idaho State University

Boston College

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Okay, So the limit of a function is as you come along the X axis as you approach, whatever the limit, it's approaching. What? What is the function approaching? So, basically, do your fingers wanna touch as you come along, The function A Z, you're closer and closer to the limit values for the first, for our first example for part A. We have the limit as ex approaches four of our function f of X. Okay, well, as X gets really, really close to four, um, the function right is approaching the value of to so limit as X approaches. Four of F X is equal to two. Okay, Now, for part B. What? We have the limit right as X approaches to now from the right. So now, instead of taking both your fingers along the function, um, you on Lee come in from the right hand side. So coming in this way from the right hand side, um, of our function. So as we is the limit as X approaches to from the right, um, of aftereffects, the function is getting really close to negative three. So the limit as X approaches to from the right off F F X is equal to negative three. How about the limit as X approaches to from the left. So now we just take our left hand and we come along to function. So we approach to now from the left while we see that as you come along from the left, the function is looking like it's getting really close to one. So the limit as X approaches to from the left of F X is equal toe one. So then, for D were asked, Um well, what is then the limit as acts approaches to Well, since we come along from the right, we approach one value, namely negative three. And as we come along from the left, we approach a different value, namely one while since these limits are not the same, the right hand limit does not equal the left hand limit. Therefore, the limit as X approaches to of F of X, um does not exist. What does this equal? Well, it doesn't even doesn't equal anything because the we do not approach one value. So therefore, the limit here as exposes two of ffx does not exist. DNA stands for does not exist. Okay, um, and then for part E. We have the limit as X approaches. Negative three now from the right. So, um, well, as we approach negative three from the right things for the function blows up. So therefore, the limit as X approaches Negative. Three from the right, we can say is equal to infinity. Um, and likewise, we approach, uh, well, So, um, for part f the limit as X approaches. Negative three now from the left off F of X. Well, we also approach, um, way approach infinity as well. So this is also equal to infinity. Um, so therefore, what is the limit as X approaches? Native three. So, Fergie, we have the limit as X approaches. Negative three. Well, since two. Both infinity here on the lemon as X approaches Negative. Three of ffx, he says, Well, equal to infinity. We could say, um how about for parts? Let's see, Part H. So our function we have the limit as ex now approaches zero from the right while we see there that the function F x is equal to infinity. And, um, what happened to be approached? My left. So, for part, I, we have the limit as X approaches zero now from the left. Well, we also blow up. We blow up in the negative direction. So therefore, the limit here as X approaches zero from the left. Well, that's gonna be equal to negative infinity. Negative infinity. Um, so then what is the limit? Same thing here since we have while you're not equal, right? If we approach, we're getting bigger and bigger as we approach from the right and we get smaller and smaller as you posting the left. So therefore right, even though they're both infinity, but one is blowing up in the positive direction was blown up in the negative direction. Therefore, the limit as X approaches zero of our function does not exist. Um, okay. And then we have what happens as we approach infinity. So this is gonna be now Part K, I guess. Here. So we have the limit as X approaches infinity of our function. Well, we can see that we're Assam tonic. Where we get closer and closer to zero as affect is bigger and bigger. So therefore, the limit as X approaches infinity of F of X is equal to zero. Okay, We never have to approach actually equals zero right for the limit. But as well as it gets bigger and bigger, the function get smaller and smaller and approach, but doesn't approach the infinitely improve approaches. Zero. Okay. And then part l we have What's the limit? As X approaches? Negative infinity. Well, now, in this case, we actually approach the value of negative one is the limit. As X approaches negative. Infinity of our function is equal to negative one. Um, yeah, and that would be I'll be it. All right.

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