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**tttcomrader** Let $\displaystyle U$ be a neighborhood of 0 in $\displaystyle \mathbb {R} $.

Suppose that $\displaystyle f:U \rightarrow \mathbb {R} $ is twice differentiable , f' and f'' are continuous, and $\displaystyle f(0)=0$.

Define $\displaystyle g: U \rightarrow \mathbb {R} $ by $\displaystyle g(x)= \frac {f(x)}{x} $ if $\displaystyle x \neq 0 $ and $\displaystyle \lim _{ h \rightarrow 0 } \frac {f(h)}{h} $ if $\displaystyle x = 0 $

Show that $\displaystyle g(0)$ exists and that g' exist and is continuous on U. Find $\displaystyle g(0),g'(0)$

Proof so far.

Find $\displaystyle g(0)$:

By the Taylor's Theorem, I can write $\displaystyle f(x)=f(0)+f'(0)x+f''(t)x^2=f'(0)x+f''(t)x^2$