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Find the values of $ a $ and $ b $ that make $ f $ continuous everywhere.

$ f(x) = \left\{ \begin{array}{ll} \dfrac{x^2 - 4}{x - 2} & \mbox{if $ x < 2 $}\\ ax^2 - bx + 3 & \mbox{if $ 2 \le x < 3 $} \\ 2x - a + b & \mbox{if $ x \ge 3 $} \end{array} \right.$

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06:29

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Boston College

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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and this problem we need to find the values of A and B. That makes this piecewise function continuous everywhere. Now to do that we have to take the limit of the function and x values where The function is described in two ways. And so this will be at X equals two and at x equals three. We're doing this because you want to make sure that the function will take on the same value as X approaches this number, either from the left or from the right. Now, the limit of the function as X approaches to from the left is equal to limit as X approaches to from the left of you have X squared minus four over x minus two, which is the same as the limit as X approaches to from the left of we have X -2 times x plus two Over X -2. this will reduce to express to and so evaluating at two, we get two Plus 2 which is four. While the limit as X approaches to from the right of f of x, this is equal to limit as X approaches to from the right, ah A x squared minus b X plus three. And so evaluating it to we get eight times 2 squared minus b times two plus three. This is just four A minus to b plus three. Now, since we want the function to be continuous then we want the limit as X approaches to of this function to exist. That means the one sided limits must be equal has. Since the one sided limits are equal then we have four a minus to B plus three. This is equal to four. Or if you simplify this, we get For a minus to be, this is equal to one. So this is our first equation for the limit of the function at ah X equals three. We have limit as X approaches three from the left of F of X. This is just limit as X approaches three from the left of A, X squared minus B. X plus three. And evaluating A three, we get eight times three squared minus B times three plus three. This is just nine A -3 B Plus three. For the limit as X approaches three from the right, we have limit As X approaches three from the right of F of X. That's just Limit as X approaches three from the right of two, X -A Plus B. That's just two times three minus A plus B. That's just 6 -5. And because you want to Make the function continuous at three, then we want the limit to exist As X approaches three and so nine A -3 B Plus three. This is equal to six -A Plus B. Simplifying this, we get 10 a -4 B. This is equal to three and this is our equation too. And so we have the system with equations For a minus to be, this is equal to one And we have 10 a -4 b. This is equal to three. Now multiply equation one by -2 and adding it to equation to, we have Have negative eight a. And then this is plus four B -2. So adding this we will get to A and this is equal to one or A is equal to 1/2. Now if a is 1/2, then substituting this either from for equation two or Question one, we get four times 1 half Minour to be, this is equal to one. You get to minus to be that's one or -2 B equals negative one and we get be equal to one half. And so these are the values of A. And B.

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