5

8.0x"O3f(-3) = litn f(z)5(-3) lim_f(z) 7_lit f(z)lim f(c) I F lim f(z)lim f()GraphGraph Df(-3) = lim f(r)in f(z)lm {(e)1 f(2)lim f()En_e)...

Question

8.0x"O3f(-3) = litn f(z)5(-3) lim_f(z) 7_lit f(z)lim f(c) I F lim f(z)lim f()GraphGraph Df(-3) = lim f(r)in f(z)lm {(e)1 f(2)lim f()En_e)

8.0x" O3 f(-3) = litn f(z) 5(-3) lim_f(z) 7_ lit f(z) lim f(c) I F lim f(z) lim f() Graph Graph D f(-3) = lim f(r) in f(z) lm {(e) 1 f(2) lim f() En_e)



Answers

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

In this problem for the function f who's graph is given to them in the following limits Right? So, here, for the part it is asked that limit extends to two for fx rate or you can see uh the limits for fx is exchange 22. Okay, so now here, in the graph you can see India graph you can see limit extends to limit Extends 2 to minus of ethics, right? Is equals to limit extends to two plus of effects. Right? So what does it mean? It is equal to zero? So therefore we can say we can easily imply it in this way that if left and limit is equal to the right hand limit then for sure the limit Exchange 2, 2 when we equals to zero. So this is a first answer. All right, now, moving ahead For the second question it has asked limit extends to three plus of FX is what? Right? It is as So we need to determine the limits. Alright, so here here the we can see yes, here it is. So here is some point where this is .5. Right? So this limits limit becomes for extends to three plus, it should be Nothing, but it should be .5 year. All right, now, moving ahead again, we are asked to find the limits of the function X. With limits extend to three minus. So here again it will be equals 2.5. Right, because only unique values present here at this .5 For this. Right, so this is actually .5 in the crap Okay, now for the part D we have limit Extent to -3 of fx is equal to what? So here we can see in the graph that is -3. So what is the valid value? Well it value is it? Okay -3. Where is the -3? So it should be uh this is uh this is standing to -3. Right? So here we can see here we have one value here it is -2. So the value comes out to be -2 here. Okay now for the part E. U. We can see zero plus zero plus is given to us Limit extents to zero plus of fx is what? So this is part E. So zero plus. Okay so what zero plus The limit should be, this is zero. Right? So for this there should be minus one. So the value for the zero plus extend to zero place of function X is equal to minus one year. Right? For zero minus again, proceed oh minus Right, so for zero minus it should be uh it should be Right. So for 0- We can see that here does not exist anything. Right? Right? So nothing is existing here. Yeah, so the left hand limit will be this will be infinitely. Okay this will be infinity. Right? So now here we can say for limit extents to zero of fx will be what? So for extents to zero it does not exist right Does not exist. Why? So, because left hand limit we have seen here for extends to zero plus an extent to my zero the values are different. It means that the value do not exist. You okay? All right. So now we can see here. One more thing. Okay, here, edge, for part edge we can see mm limit extends to infinity of fx is what support for infinity it will be. It will be same as infinity only. Right? And for minus infinity it will be zero. Okay, because this cuts the region right? This is cutting the region, right? So that's where 20. But this is infinity because we are getting this infinity in this form. Right? So it is not cutting the uh X axis at any point, right? At X tends to infinity. Right? But in minus infinity. Just starting to uh X axis. That means the value becomes zero. So this is how we solve this problem. I hope you understood the concept. Thank you for watching

Mhm. Hello. So here for part A we have the limit as X approaches negative three of ffx. So we can see from observing the figure that um as the function approaches a value of two from the left and from the right, when X approaches negative three. Um The limit as X approaches negative three is then too. So we see here the limit as X approaches negative three of the function F X from the given figure is going to be equal to two. Um Then for B uh we have f of negative three. So F of negative three. Um Well the value of the function at X is when when what X is negative three. We can see from the um the figure that the value of the function there is one so therefore Um of -3 is just equal to one. Mhm Than 4? Part C of -1. Well, if we look at um so this is not eliminated more. This is the value of the function um When the input is negative one and when actions negative one, the function is not defined. So F of negative one um does not exist. We could say or is not defined so does not exist. You need for does not exist. Um And then for party we have the limit as X approaches -1. So the limits as X approaches negative one of the function F of X. That is gonna be equal to. Well again, we can see from observing the figure that the function approaches a value of 2.5 from the left and from the right as X approaches negative one. So therefore we say the limit As X approaches -1 of our function is equal to Uh 2.5. And for part E. Um F of one. Well what is the value of the function when access equal to one? We can see the value. The function when x is one is to. So therefore we say that f of one is equal to two. Yeah. And for part F we have the limit as X approaches one. So well, so when X is one right, the value is too, but the limit doesn't really care about what happens um when X is one because what happens what's really really close to one? So the limit as X approaches one of fx? Well, we can see from the figure that the function approaches a value of two from the left, but a value of one from the right. And since these values do not match, we conclude that there is no limit when X um for when X approaches one of the limit, as X approaches one of Quebec's does not exist because we approach to different values on the left and from the right and then um part G we have of the limit as X approaches one. Now this little minus sign here means you read this is the limit as X approaches one from the left and it's only on one side of limit. So the one sided limit here as X approaches one from the left is too, so the limit as X approaches one from the left, That is equal to two. Um and then if you put a little instead of a mind inside the plus side that would be approaching from the right. So for part G part H we have the limit as X approaches one from the right. Well that one sided limit um is as X approaches one from the right. Um the function approaches one. The limit as X approaches one from the right is is equal 21 And for part II we have the limit as X approaches negative one from the right to the limit. The limit as X approaches um negative one from the right. Well, again, this is a one sided limit, and as X approaches negative one from the right, the function approaches 2.5. The limit as X approaches negative one from the right is equal to 2.5. All right, all of them, yeah.

All right. So we have some limits to evaluate from the scratch. So, eh, the limit and expertise for F of X. So we're looking and yeah, from the left and from the right. Looks like approaching too. And then being the king limit is expert just two from the right. So given the right, we're going down to the whole of negative three. Seeing good two from the left. Now we're going from that ass until zero from down to Teo are stopping one okay. And see when his ex approaches to. Because the left and the right limits don't equal does not exist, even me. Okay, for e dilemma is extra purchase negative Three from the right. So we haven't asked himto X equals negative three and push positive insanity in both directions. So half its negative three from the left. So so approaching positive infinity and then the limit as except britches. Negative three. Well, because from the left and the right, it's both going to positive infinity. We can conclude that the women so virtuous minds three is positive. Infinity. However, for h, the limit is X approaches Zero from them, right is going to positive infinity and I from the left. It's going toe minus and felony. That means the limit is X approaches. Zero does not exist because fromthe left, it's approaching minus infinity and from the rights approaching possible vanity. And then l lim is extra purchase minus infinity. Well, it looks like from the graph, I'm skipping Kay. Okay, now. Oh, okay. So for infinity, it's X goes to infinity and looks like the graph is leveling off to zero. And then it's experience. Mind that infinity for leveling off to negative one.

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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