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This exercise illustrates how derivatives may be used to compute limits in one very special and important case. Consider $\lim _{x \rightarrow a} \frac{f(x)}{g(x)},$ where $f$ and $g$ are differentiable functions with $f(a)=g(a)=0$. (a) Show that $$\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{f^{\prime}(a)}{g^{\prime}(a)}$$ provided $g^{\prime}(a) \neq 0 .$ Hint: Use the alternate definition of the derivative as given in Exercise 37 in Section $3.1 .$ (b) Show that if the derivatives of both $f$ and $g$ are continuous functions then the above result may be written as $$\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}$$ This result is a special case of a theorem known as L'Hôpital's rule.

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 5

Derivative Rules 2

Derivatives

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

01:29

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16:58

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eso. The way this problem works out is getting you to recognize that limits. X approaches. A of ffx over G of X is equal to each individual limit limit as X approaches A of f of X over G of X. Um, and what we can do is we can actually break down. Uh, yeah, the alternative definition of the derivative. You know, if we were to write out F prime of X is equal to the limit as X approaches a off f of X minus f obey over X minus A. If I did the same thing for G of X Hey, planet X limit as X approaches a of G of X minus G ave over X minus A. Well, if we're going to divide each piece individually, what will end up with us? It's just dividing individually and, um, getting rid of these equal science. What we end up with is multiplying by the reciprocal of this. So what we have is the limit as X approaches A after that's minus f today over X minus a help. Put some qualities in their times. Uh, X minus a over. You have ax minus G r a and we could cancel out some pieces. And what we're left with is essentially this piece right here. As long as the limits X approaches A GF explains Juve is equal to some non zero constant. That is sort of important in here due to, you know, can't divide by zero. Um, which brings us to the rule that will establish later on which is local Taliban rule. Uh, help you understand you are a new method Might be a better way of saying, um that the derivative, um well, let me straight out that a limit as X approaches a of ffx over G of X is equal to a limit as X approaches a of f prime of X off the G prime events.

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