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Problem 83 Hard Difficulty

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

Answer

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

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Video Transcript

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