Compute $\lim\limits_{x\to0}\left(\frac{1}{x^4}-\frac{\int_0^{x^2}e^{-u^2}du}{x^6}\right)$ without L'Hopital rule nor Taylor

Question

I would like to compute the following $$\lim_{x\to0}\left(\frac{1}{x^4}-\frac{\displaystyle\int_0^{x^2}e^{-u^2}du}{x^6}\right)$$

I have done it using L'Hospital rule and Taylor and I obtained $1/3$

Is there a way I can compute this limit without appealing L'Hospital rule or Taylor? I am likely thinking about squeezing or a more elegant way.

Answer 1

The expression under limit can be written as $$\frac{1}{x^6}\int_{0}^{x^2}(1-e^{-u^2})\,du$$ We use the standard inequality $$1+t\leq e^t\leq \frac{1}{1-t},0\leq t <1$$ and take reciprocal to get $$1-t\leq e^{-t} \leq\frac{1}{1+t} $$ Putting $t=u^2$ where $0\leq u\leq x^2<1$ we get $$1-\frac{1}{1+u^2}\leq 1-e^{-u^2}\leq 1-(1-u^2)$$ ie $$\frac{u^2}{1+u^2}\leq 1-e^{-u^2}\leq u^2$$ Since $0\leq u\leq x^2$ we have $$\frac{u^2}{1+x^4}\leq 1-e^{-u^2}\leq u^2$$ and integrating with respect to $u$ on interval $[0,x^2]$ we get $$\frac{x^6}{3(1+x^4)}\leq \int_{0}^{x^2}(1-e^{-u^2})\,du\leq \frac{x^6}{3}$$ By Squeeze theorem the desired limit is $1/3$.

Answer 2

with $t^2=u $ the integral becomes

$$2\int_0^x e^{-t^4}tdt $$ $$=2\int_0^x (t-t^5+t^5\epsilon (t))dt $$ $$=x^2-\frac {x^6}{3}+x^6\epsilon (x) $$

the limit will be $$\frac {1}{3} $$