Question

Suppose $f\left(x\right)$ is differentiable at $x=1$. If $f\left(1\right)=0$ and $\underset{h\to 0}{lim}\frac{1}{h}f(1+h)=5$, then $f^{\prime }\left(1\right)$ equals

• A. $6$
• B. $5$
• C. $4$
• D. $3$

Answer 1

Answer: (B)

$5$

Given that $\underset{h\to 0}{lim}\frac{f(1+h)}{h}=5$
When $h\to 0$, both the numerator and denominator reduce to 0. Hence the fraction is in $0/0$ form. Also given that $f\left(x\right)$ is differentiable at $x=1$. So we can also use L'hopitals rule to find limit. Taking derivative of both denominator and numerator we get,
$\underset{h\to 0}{lim}\frac{f(1+h)}{h}=\underset{h\to 0}{lim}\frac{f^{\prime }(1+h)}{1}=f^{\prime }\left(1\right)$
Hence $f^{\prime }\left(1\right)=5$