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Evaluate $ \lim_{x\to1} \frac {x^{1000} - 1}{x - 1}. $

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$\lim _{x \rightarrow 1} \frac{x^{1000}-1}{x-1}=\lim _{x \rightarrow 1} \frac{(x-1)\left(x^{999}+x^{998}+x^{997}+\cdots+x^{2}+x+1\right)}{x-1}=\lim _{x \rightarrow 1}\left(x^{999}+x^{998}+x^{997}+\cdots+x^{2}+x+1\right)$ $=\underbrace{1+1+1+\cdots+1+1+1}_{1000 \text { ones }}=1000,$ as above.

01:05

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Derivatives

Differentiation

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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it's clear it's the one you married here. We're going to evaluate the limit as X approaches one for X to the 1000 power minus one over X minus one. So we're gonna write this as a derivative of some function. So when we rewrite the given limit, we get this becomes equal to the limit. Has X approaches one for X to 1000. Power minus one to the 1000 power over. Sex minus one. This is equal to the derivative aren't one. And this is FX is equal to X to the 1000 power. We're gonna use the power roll so we get. Our derivative is equal to 1000 next to the 1000 minus one. This is 999. So are derivative at one is equal to 1000. So the limit as X approaches one for X. The 1000 power minus one over X minus one is equal to the derivative. That one which is equal to 1000

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