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DM
Numerade Educator

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Problem 13 Easy Difficulty

Use the graph of the function $ f $ to state the value of each limit, if it exists. If it does not exist, explain why.
(a) $ \displaystyle \lim_{x \to 0^-}f(x) $
(b) $ \displaystyle \lim_{x \to 0^+}f(x) $
(c) $ \displaystyle \lim_{x \to 0}f(x) $

$ \displaystyle f(x) = \frac{1}{1+e^{1/x}} $

Answer

(a) $\lim _{x \rightarrow 0^{-}} f(x)=1$
(b) $\lim _{x \rightarrow 0^{+}} f(x)=0$
(c) $\lim _{x \rightarrow 0} f(x)$ does not exist because the limits in part (a) and part (b) are not equal.

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Video Transcript

So in this problem we are asked to find some limits of function F of X equals 1/1 plus E. To the one over X. Let's look at this graph for just a second here, this is the graph of it Coming in from the left. Notice it goes up to the value of one And for the right starts down here at zero and trails up like that accordingly. Okay, so the first one we're asked is the limit As X approaches zero from the left F of X. Well, as we just saw coming in from the left here, I'm over here and I'm coming in from the left. What happens? Well, I come up to the value of one, don't I? This is one. So the limit as X approaches zero from the right of F of X. Well, I'm over here at the right and trail in towards zero. As you see there becomes a value of zero, doesn't it? Zero. And then we're asked to limit as X approaches zero of F of X. And the problem we have is that these two limits here are not equal. And since they're not equal, the limit from the left and limit from the right and that means the limit at that value of X does not exist because if it did exist, then limit from the left and the limit from the right would be equal