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For what value of the constant $c$ is the function $f$ continu-ous on $(-\infty, \infty) ?$$$f(x)=\left\{\begin{array}{ll}{c x^{2}+2 x} & {\text { if } x<2} \\ {x^{3}-c x} & {\text { if } x \geqslant 2}\end{array}\right.$$

$$c=\frac{2}{3}$$

Calculus 1 / AB

Chapter 1

FUNCTIONS AND LIMITS

Section 5

Continuity

Functions

Limits

Continuous Functions

Campbell University

Harvey Mudd College

Baylor University

University of Nottingham

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Well, let's no cities function. Flags equal. It's equal to feet. Ex queer plus two eggs you access is less than two. Oh, ex queued My no see eggs if X is greater or equal to and see is just and constant. So let's try to work out the value off this constant for the function to be continuous. So we want dysfunction to be continuous in on minus infinitive plus and fill it so everywhere for any really valuable backs. Andi were to work out the value off this constant, which makes the function continuous everywhere. So does let's use the definition of continuity. So first the world well, these are just born in no bills, so they are continuous everywhere for any real value. Kovacs. Now the only issue is at two when x equal to this is the only point where the function may up discontinue tive. So in order to get a continuous function at X equal to the usual requirement, af two has to be equal to the limit. Well X, a purchase to of the function. So I fought to the function at X equal to well, it's equal. We see when x equal to the function is X cubed minus asiax So it will be X cubed mills to Q minus to see. That's the value off I forgot to. We can rewrite it as eight miners to see No, what about the limit? Well, the limit of the function when eggs and porches to from the left is equal to when the function. In this case, the function we have to choose is the 1st 1 see X squared plus two acts. And when X supporters do, this is a continuous function. So what can we write? What we just plug in the valley to So we're going where going to on with four C plus four that said the limit when x purchased two from the ride. Well, in this case to limit as going to roar this function execute minus see eggs because X is greater than two on. And what about this payment? Well, this is going to be two to the power off free once more minus two c. So is just eight minus two c, which is exactly I fall too. No, If we want Thies to limits to be the same well, what should we solve? just the question for C plus for the limit from the left has to be equal to the limit from the right a minus two C So that's orbit for C means for C class to see Ikhwan a minus four mean six C equal for C equal for over six. We can simplify. We get to over free. Thes is the volume off. See according to which this function is containers everywhere on minus infinity on the interval, minus infinity plus infinity.

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