Suppose f(x) is differentiable at x=1 and _h01hf(1+h)=5, then f'(1) equals

Question

Suppose $f(x)$ is differentiable at $x=1$ and $\displaystyle\lim_{h	o0}\dfrac1hf(1+h)=5$, then $f'(1)$ equals

Accepted Answer

Since $f$ is differentiable (hence continuous) at $1$, and $\displaystyle\lim_{h	o0}\dfrac{f(1+h)}{h}=5$ finite, we must have $f(1)=0$. Then $f'(1)=\displaystyle\lim_{h	o0}\dfrac{f(1+h)-f(1)}{h}=\displaystyle\lim_{h	o0}\dfrac{f(1+h)}{h}=5$.

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