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If $ r $ is a rational function, use Exercise 57 to show that $ \displaystyle \lim_{x \to a}r(x) = r(a) $ for every number $ a $ in the domain of $ r $.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

University of Michigan - Ann Arbor

Idaho State University

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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If $r$ is a rational funct…

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If the domain of a rationa…

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In Exercises $47-56,$ find…

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Prove that if the number $…

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Use an algebraic simplific…

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This is problem number fifty eight out just to recap Cassie's edition section two point three. If our is a rational function, use Exercise fifty seven to show that filament is expert to say of our vics is equal to are away for every number aimed that Amenabar. So the result from Exercise fifty seven is that the limit and six approaches a of a polynomial p is equal to the polynomial evaluated at a so let's work with irrational function. And let's consider that this rational function can be represented as a ratio of two polynomial. Lt's P and Q King. So element his ex opportunity of this rational function are is the same as the limited expert dizzy of a polynomial peed right by upon a meal. Q. In order to use the following Lim alone, we need to be certain as he approached. A Cuba does not equal zero. Basically, we can't have this be true, and in those long as that is true, a song is a cute is not equal zero for any valuing. Then we can use the Lim lock for division to have the limit. That's X approaches a lot of people divided by the limit is expressing of Q and Cincy to these upon a male's separately. The blame it on the top equals upon a well P rally today. And the pond and one on the bottom equals kill Levi would have value, eh? And this ratio, if here accuses him, is this rational function so of its value It'd eh Then this gives us the rational function devaluated, eh? And we have shown that for everybody A in the domain, a bar where the denominator does not equal zero, that this stands for all rational functions under these conditions.

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