Given-that-lim-x-0-f-x-x-h-L-then-lim-x-0-f-x-2x-h-

Question Number 80172 by Rio Michael last updated on 31/Jan/20

G i v e n t h a t \lim_{x \to 0} \frac{\sqrt{f (x) + x}}{h} = L t h e n

\lim_{x \to 0} \frac{\sqrt{f (x) + 2 x}}{h} = ?

Commented by john santu last updated on 31/Jan/20

\lim_{x \to 0} \frac{\sqrt{f (x) + x}}{h} o r \lim_{x \to 0} \frac{\sqrt{f (x) + x}}{x} ?

Commented by john santu last updated on 31/Jan/20

w e a s s u m e

\lim_{x \to 0} \frac{\sqrt{f (x) + x}}{x} = \lim_{x \to 0} \sqrt{\frac{f (x) + x}{x^{2}}} = L

\sqrt{\lim_{x \to 0} \frac{f (x) + x}{x^{2}}} = L

\lim_{x \to 0} \frac{f^{'} (x) + 1}{2 x} = L^{2}, (f^{'} (0) = - 1)

\lim_{x \to 0} \frac{f^{″} (x)}{2} = L^{2} \Rightarrow f^{″} (0) = 2 L^{2}

s i m i l a r l y \lim_{x \to 0} \frac{\sqrt{f (x) + 2 x}}{x} = L