If $f(x)$ is differentiable function and $f^{''}(0) = a$, then $\lim_{x \rightarrow 0}\frac{2f(x) - 3f(2x) + f(4x)}{x^{2}}$ is equal to

Question

Dr. Abubacker · Accepted Answer

(a)
$\lim_{x \rightarrow 0}\frac{2f(x) - 3f(2x) + f(4x)}{x^{2}}$
$= \lim_{x \rightarrow 0}\frac{2f'(x) - 6f'(2x) + 4f'(4x)}{2x}$
$= \lim_{x \rightarrow 0}\frac{2f''(x) - 12f''(2x) + 16f''(4x)}{2}$
$= \frac{2f^{''}(0) - 12f^{''}(0) + 16f^{''}(0)}{2} = \frac{6a}{2} = 3a$

Q. If $f(x)$ is differentiable function and $f^{''}(0) = a$, then $\lim_{x \rightarrow 0}\frac{2f(x) - 3f(2x) + f(4x)}{x^{2}}$ is equal to

Explanation

Reviewed by Dr. Abubacker

Want more practice?