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Here is a well-known function:

$$f(x)=\begin{cases}\exp\left(-\frac{1}{x^2}\right) & x\not=0 \\ 0& x=0\end{cases}.$$

How to calculate :

$$\lim_{t \rightarrow 0}\frac{f(t^2+t)-f(t)}{t^2},$$ does it equal zero?

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have you tried power series or taylor expansion on f(x) when x is not 0? – Enzo Mar 6 '13 at 14:13
up vote 2 down vote accepted

Check if $f'(0),f''(0)$ exists or not using standard definition and then apply L'Hopital's Rule.

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Your function is the typical example of a function of class $C^\infty$ whose derivatives of any order are zero at $x=0$. This implies easily that your limit is zero too.

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