Find $f ' (0)$ for the function $f(x) = g(x)/x^2$ when $x\not = 0$ and $ f(x)=0$ for $x=0$

Question

Find $f '(0)$ for the function \begin{align} f(x)=\begin{cases} \frac{g(x)}{x^2}, & \text{if }x \not = 0\\ 0, & \text{if }x=0 \end{cases} \end{align}

With \begin{align} g(0)=g'(0) = g''(0) = 0 \\ g'''(0) = 14 \end{align}

score 3 · Answer 1 · answered Apr 10 '13 at 06:52

3

Hint: Think of $$f'(0)=\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}$$

answered Apr 10 '13 at 06:52

Mikasa

Nice hint - my you've grown an impressive beard :-) +1! – Amzoti Apr 13 '13 at 05:12
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@Amzoti: He is Gandalf. Gandalf is a character and you know him if you see the movie called Hoobit! – Mikasa Apr 14 '13 at 05:10
1

@BabakS.: Yes, I actually like him as he has excellent wisdom, street smarts and knows just how much to help! The video comes out soon of the latest movie. Hope all is well and good to see you around as usual! – Amzoti Apr 14 '13 at 05:12

P.. · Answer 2 · 2013-04-10T19:03:04.497

0

Hint: Use the definition and L'Hôpital's rule.

The answer is $\dfrac pq$ with $p,p-q\in\{7,1,8,6,4\}$.

edited Apr 10 '13 at 19:03

answered Apr 10 '13 at 06:59

P..

Are you going to change the numbers if your reputation increases ;) ? Or is 7186 a coincident ? – Thomas Apr 10 '13 at 07:04
@macydanim :) Actually is for user71864. I didn't see the connection with my reputation. Further hint: If my reputation increase to 7194 I will change it yes. – P.. Apr 10 '13 at 07:08
Oh I didn't see the user ID matches the numbers as well. Scary! – Thomas Apr 10 '13 at 07:10
@macydanim: Do you know the strong law of small numbers? – P.. Apr 10 '13 at 07:12
No I didn't know that. Thanks for that. Though 7186 isn't that small ;) – Thomas Apr 10 '13 at 07:16

2 Answers2