Find convergence and limit of $\lim_{x\to0}\frac{e^{-x^2}-(1+ax^2)}{x^4}$

Question

I have an exercise of finding a real number "a" and limit of $$\lim_{x\to0}\frac{e^{-x^2}-(1+ax^2)}{x^4}$$

I have no idea what should I do, can someone help me solve this step-by-step? Thank you in advance.

https://math.stackexchange.com/questions/387333/are-all-limits-solvable-without-lhôpital-rule-or-series-expansion — lab bhattacharjee, Apr 13 '19 at 13:47

score 4 · Accepted Answer · answered Apr 13 '19 at 13:19

4

Hint: Use the fact that$$e^{-x^2}=1-x^2+\frac{x^4}2-\frac{x^6}{3!}+\cdots$$

answered Apr 13 '19 at 13:19

Thanks for the hint. But how should I apply that? – S.Ky Apr 13 '19 at 13:45
You should apply it in order to show that there is one and only one $a$ for which that limit exists: $a=-1$. – José Carlos Santos Apr 13 '19 at 13:46

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