Question

Let $f ( x )$ be a twice differentiable function and $f ''(0)=5$, then the value of $\displaystyle\lim _{x \rightarrow 0} \frac{3 f(x)-4 f(3 x)+f(9 x)}{x^{2}}$ is

• A. 0
• B. 120
• C. -120
• D. does not exist

Answer 1

Answer: (B)

$\displaystyle\lim _{x \rightarrow 0} \frac{3 f(x)-4 f(3 x)+f(9 x)}{x^{2}} \,\,\,\,\left(\frac{0}{0}\right.$ form $)$
$=\displaystyle\lim _{x \rightarrow 0} \frac{3 f'(x)-12 f'(3 x)+9 f'(9 x)}{2 x}\left(\frac{0}{0}\right.$ form $)$
$=\displaystyle\lim _{x \rightarrow 0} \frac{3 f''(x)-36 f''(3 x)+81 f''(9 x)}{2}$
$=\frac{3 f''(0)-36 f''(0)+81 f''(0)}{2}$
$=24 f'(0)=24.5=120$