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  • Evaluate the limit      <img alt="\mathrm{\lim _{x \rightarrow 0} \frac{e^{-\frac{1}{x^2}}}{x^2}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7B

Evaluate the limit      \mathrm{\lim _{x \rightarrow 0} \frac{e^{-\frac{1}{x^2}}}{x^2}}\mathrm{\lim _{x \rightarrow 0} \frac{e^{-\frac{1}{x^2}}}{x^2}}

Option: 1

0


Option: 2

1


Option: 3

-1


Option: 4

2


Answers (1)

best_answer

If you must use L'Hospital's Rule, here is one way forward. Note that  \mathrm{\frac{e^{-1 / x^2}}{x^2}=\frac{1 / x^2}{e^{1 / x^2}} } and that   \mathrm{\lim _{x \rightarrow 0} e^{1 / x^2}=\infty }  Then, we can write
\mathrm{\lim _{x \rightarrow 0} \frac{e^{-1 / x^2}}{x^2} =\lim _{x \rightarrow 0} \frac{1 / x^2}{e^{1 / x^2}}}
                \mathrm{\overbrace{=}^{L H R} \frac{\left(-2 / x^3\right)}{\left(-2 / x^3\right) e^{1 / x^2}} }
                   \mathrm{ =\lim _{x \rightarrow 0} \frac{1}{e^{1 / x^2}} }
                   =0.
 

Posted by

Divya Prakash Singh

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