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Explain why the function is discontinuous at the given number $ a $. Sketch the graph of the function.

$ f(x) = \left\{ \begin{array}{ll} \dfrac{1}{x + 2} & \mbox{if $ x \neq -2 $} \hspace{40mm} a = -2\\ 1 & \mbox{if $ x = -2 $} \end{array} \right.$

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02:31

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Harvey Mudd College

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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$17-22$ Explain why the fu…

we have a question in this um Effects is being given faxes one by X plus two. If X is not equal to -2 and one of annex equal to -2 we have to say that we have to explain that but the function is just continuous at Why it is discontinuous set actually equal to -2. Okay, no, to get the discount to get continue to test, let us finds first left and limit to limit X approaches to -2 from left side. So this will come here which will be called to one by X-us two. So simply since this is approaching two from left side, so this has to be positive. So plus infinity. Alright and limit limit X approaches to -2 from right side -2 will be approaching -2 from right side which means ah This is -2 and this is to so if we are approaching this from here from right side Okay we have a protein -2 from right side -2 from but they said so this will be of one by okay minus two from right side so it will be better than minus two. X will be somewhere better than minus two. Um let us say minus one, It was less than -2 the same -3 so it has to be negative. Sorry for this negative explains to it has to be bless infinity. Okay, no uh the limit value of this function at X equal to minus two is one so we can see that left and limit is not equal to right and limit is not equal to value this function at X equal to a. So this continuous discontinuous at X equal to -2. We have to draw the graph. So for that to try the graph we had to just find out some various things affects equal to one by X. Plus two. For this this is But that's not equal to -2. Of course. Okay, so for this we have to first find the X intercept and it is easy that it's equal to minus two. Is the S in total? Okay, X equal to minus two. Is there some toad? Now X intercept for X intercept effects will be zero. So one will be equal to zero. So no X intercept. Okay, for my intercept for my interest in we're plugging in X equal to zero. So why intercepted zero comma one by two and X equal to minus two. Is the vertical isn't taught. And here since the degree of numerator is lesser than the denominator so why call to zero that his ex success will be the horizontal symptoms. Okay, so let us draw the graph And yes, no problem. We will be trying the gulf. This is X. This is why x equal to -2. So this is the vertical lesson. Tote particular and horizontal isn't Ortiz act success. And uh why intercepted zero comma one by two. This is X equal to minus students will come out one by to let us say this is zero comma I want to buy two. So our graph would look like this. This is for all the values of a All the village of ex except x equal to -2 and it is one attacks equal to -2. So this is If this is one so If X equal to -2, this is what, so this is the graph And we can easily find out that if we are approaching -2 from left side, so it goes towards minus infinity and if we are approaching -2 from right side it goes towards bless infinity. So yes, all our kind. Thank you so much.

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