Evaluate $\underset{x\to \mathrm{\infty}}{lim}{e}^{-{x}^{2}}{\int}_{x}^{x+\mathrm{ln}(x)/x}{e}^{-{t}^{2}}dt$

antennense
2022-07-13
Answered

Evaluate $\underset{x\to \mathrm{\infty}}{lim}{e}^{-{x}^{2}}{\int}_{x}^{x+\mathrm{ln}(x)/x}{e}^{-{t}^{2}}dt$

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asked 2022-07-01

I have to evaluate the following without L'Hopital's rule

$\underset{x\to \mathrm{\infty}}{lim}x\mathrm{tan}(1/x)$

I can simplify this to be

$\underset{x\to \mathrm{\infty}}{lim}x\mathrm{sin}(1/x)$

because

$\underset{x\to \mathrm{\infty}}{lim}\mathrm{cos}(1/x)=1$

However, after that, I'm totally lost. L'Hopital's rule seems like my only option. Can someone help me out?

$\underset{x\to \mathrm{\infty}}{lim}x\mathrm{tan}(1/x)$

I can simplify this to be

$\underset{x\to \mathrm{\infty}}{lim}x\mathrm{sin}(1/x)$

because

$\underset{x\to \mathrm{\infty}}{lim}\mathrm{cos}(1/x)=1$

However, after that, I'm totally lost. L'Hopital's rule seems like my only option. Can someone help me out?

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