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Given the following limit,

$$\begin{align}

\lim_{x\to 0}\frac{e^{-1/x^2}-0}{x-0}\\\\

&

\end{align}$$

How do I calculate it? when pluggin in 0 I would get $\frac{0}{0}$

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The problem here is that applying L’Hôpital’s rule right of the bat yields

$$\begin{align}

\lim_{x\to 0}\frac{e^{-1/x^2}-0}{x-0}&=\lim_{x\to 0}\frac{2e^{-1/x^2}}{x^3},

\end{align}$$

which isn’t particularly helpful. However, we notice that

$$\begin{align}

\lim_{x\to 0^+}\frac{e^{-1/x^2}-0}{x-0}&=\lim_{x\to 0^+}\frac1{xe^{1/x^2}}\\

&=\lim_{x\to\infty}\frac x{e^{x^2}}\\

&=\lim_{x\to\infty}\frac1{2xe^{x^2}}\\

&=0;

\end{align}$$

similar manipulations can be used to confirm that the left-handed limit is zero as well. Thus

$$\begin{align}

\lim_{x\to 0}\frac{e^{-1/x^2}-0}{x-0}&=0.

\end{align}$$

We don’t need to use L’Hospital’s Rule here.

Simply note that $e^x\ge 1+x$, which I proved in This Answer. Hence, we have

$$\left|\frac{e^{-1/x^2}}{x}\right|=\left|\frac{1}{xe^{1/x^2}}\right|\le \frac{|x|}{1+x^2}$$

whereupon taking the limit yields

$$\lim_{x\to 0}\frac{e^{-1/x^2}}{x}=0$$

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