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Sketch the graph of the function and use it to determine the values of $ a $ for which $ \displaystyle \lim_{x\to a}f(x) $ exists.$$ f(x) = \left\{ \begin{array}{ll} 1 + x & \mbox {if $ x < -1 $}\\ x^2 & \mbox{if $ -1 \le x < 1$}\\ 2 - x & \mbox{if $ x \ge 1 $} \end{array} \right.$$

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05:47

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 2

The Limit of a Function

Limits

Derivatives

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Idaho State University

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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Sketch the graph of the fu…

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Sketch the graph of the fo…

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defined in three different parts. Uh function is defined to be one plus X. When x is less than negative one, that's the red portion of the graph that you see. The function f of X equals X squared. When uh negative one is less than or equal to ex uh ex in turn is strictly less than one. That's the blue portion of the graph that you see. And last but not least when x is greater than or equal to one. Uh F of X is defined to be two minus X. We want to find all values a along the X axis for which the limit exists. And this graph is uh going to greatly help us. You can see um that this uh function is basically connected. There's no breaks anywhere in the function except right here. When X is negative one, we have a break into function when X is positive one. Uh You can see the blue portion of the function uh meets nicely uh the green portion of the function. And so the limit as X approaches one from the negative side, the blue side is going to be the same limit as X approaches one from the uh positive side. The green side. But over here, it when uh X is negative one, or specifically when a is equal to negative one. Uh This function uh does not have a limit in other words. Uh The limit of f of X as X approaches negative one does not exist. Um The limits exist for dysfunction for any other value, any other real number value along the X axis, it does not exist. The limit of dysfunction does not exist when X approaches the value of negative one because as X approaches negative one from the negative side to function uh approaches zero. So the limit of F of X as X approaches negative one from the negative side is zero, but the limit of F of X pay attention to the blue portion of the curve. Now the limit of F of X as X approaches negative one from the positive side, uh F of X is approaching the value of one. And so the limit of F of X as X approaches negative one does not exist because the limit of F of X as X approaches negative one from the negative side does not equal the limit of F. Of X as X approaches negative one from the positive side. And you can clearly see that because the red portion of the graph is getting close to zero, think of the Y axis as to function values, whereas the blue portion of the graph is getting close to the value of one

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