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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}} $

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(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.

02:11

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Limits

Derivatives

Campbell University

Oregon State University

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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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

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