Limit, continuity, derivative, and integral

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Problem

For $x\ne1$ , let $f(x)=(x^2-1)/(x-1)$ . Choose $f(1)$ so that $f$ is continuous. For that continuous extension, find $f'(1)$ and $\int_0^2 f(x)\,dx$ .

Reveal answer or reference solution

For $x\ne1$ , $f(x)=x+1$ , so $\lim_{x\to1}f(x)=2$ . Set $f(1)=2$ . The extended function is $x+1$ everywhere, hence $f'(1)=1$ and

\int_0^2(x+1)\,dx=\left[\frac{x^2}{2}+x\right]_0^2=4

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Limit, continuity, derivative, and integral

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