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Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{(3 + h)^{-1} - 3^{-1}}{h} $

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$-\frac{1}{9}$

02:24

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Missouri State University

Oregon State University

Harvey Mudd College

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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suppose you want to evaluate the limits of three plus age. Race two negative one minus three. Race the negative one over age as H approaches zero know that if we directly substitute H equals zero, we have three plus zero. Race two negative one minus three. Race the negative 1/0 which is just three. Race two negative one minus three. Race. The negative one over zero or that's just 0/0. Now this is indeterminate. That means we need to rewrite our function. And so from here we have the limit as H approaches zero of not that we can rewrite three plus age raised negative one and two one over three plus H minus the same way we have for three. Race two negative one that will be 1/3. This all over H. And then we combine the numerator. We get limit as H approaches zero of three minus three plus H over three times three plus age. This times the reciprocal of a trick is won over age. And then from here we simplify and we get the limit as a jew approaches zero of three minus three minus H over three times three plus age. This times one over age. And we get limit as H approaches zero of negative age over three times three plus age times one over age and know that we can cancel at the beach. And we have limit as H approaches zero of the negative one over three times three plus age and so substituting zero for age we have negative one over three times 340 This is just negative 1/9. And so this is the value of the limits.

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