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The graphs of $ f $ and $ g $ are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why.

(a) $ \displaystyle \lim_{x \to 2}[f(x) + g(x)] $

(b) $ \displaystyle \lim_{x \to 0}[f(x) - g(x)] $

(c) $ \displaystyle \lim_{x \to -1}[f(x) g(x)] $

(d) $ \displaystyle \lim_{x \to 3}\frac{f(x)}{g(x)} $

(e) $ \displaystyle \lim_{x \to 2}[x^2 f(x)] $

(f) $ \displaystyle f(-1) + \lim_{x \to -1}g(x) $

a. $\lim _{x \rightarrow 2}[f(x)+g(x)]=1$

b. $\lim _{x \rightarrow 0}[f(x)-g(x)]$ does not exist

c. $\lim _{x \rightarrow-1}[f(x) g(x)]=2$

d. $\lim _{x \rightarrow 3} \frac{f(x)}{g(x)}$ does not exist

e. $\lim _{x \rightarrow 2}\left[x^{2} f(x)\right]=-4$

f. $f(-1)+\lim _{x \rightarrow-1} g(x)=5$

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So in this problem were asked were given these functions F and G we're asked to find a number of limits. So the first one is the limit as X approaches to of F of X plus G F X. Okay, bye limit laws. This is the same as the limit as X approaches to F of X plus the limit as X approaches two of G F X. Okay, so on F Mhm. The limit as X approaches to. Right, well, His ex approaches to this goes towards -1, doesn't it? No matter which direction we come from. Okay, so that's -1 plus on G as X approaches to as we come in here and no matter which direction we come from, G is too. So this is one. Okay? He says the limit as X approaches zero, uh Ffx minus G F X. Well, again, This would be the limit as X approaches zero of F of X plus Limit as X approaches zero of G of X. By limit laws. and as X approaches zero over here on F. Well, that goes to too, doesn't it? But what happens on G G is discontinuous at X equals zero because we go there we go there depending on if we're coming from the left or from the right, well, this does not exist. Therefore this limit in total does not exist. All right, seen the limit as X approaches minus one um F of X G of X. All right, so let's see something here for a minute As X approaches one on F. That's discontinuous, isn't it? Well, actually They both come in toward one and what happens on G F X. Well They come into two. Okay, So this is to one times 2 which is two. All right, so on D we have the limit, It was x approaches three of half of X over D F X. This X approaches three. So let's look up here for a minute. 123, SSF of X Approaches three. Then our limit is two up there And his ex approaches three on GFX. Well That goes to zero and now I have a problem because I cannot Divide by zero so this does not exist. All right. He says the limit as X approaches two of X squared F of X. All right, well, let's see F of X as X approaches two is what X approaches to? Well, that's minus one and then two in there for X squared is four, so this is minus four, isn't it? Okay. Yeah. Last one Half at -1. Was limit As X approaches -1 of G F X. Okay, let's see F -1. Put -1 in here. F -1. That's all the way up here at this point, That's 1, 2, 3, that's it. three. So it was three plus the limit as X approaches modest one of G of X. Well, that is to right because it's right here, so that's two. So this is five And there we go. Have all five answers.

Oklahoma State University