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I need to solve this limit without using L'Hospital's rule.

$$\lim_{x\to 0}⁡(\cos x)^\left(\frac{-4}{x^2}\right)⁡$$

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closed as off-topic by TomGrubb, RRL, Scientifica, Shailesh, Leucippus Nov 14 '18 at 0:35

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  • $\begingroup$ This looks like a homework problem. Have you tried anything? $\endgroup$ – Robert Thingum Nov 13 '18 at 20:23
  • $\begingroup$ I have solved it for a friend using l'Hospital's rule. Unfortunately, my friend is attending high school and he doesn't know this rule. I don't remember how to solve it without using l'Hospital's rule.That's why I am asking for an advice here. $\endgroup$ – Alexandra Nov 13 '18 at 20:27
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Hint:

$$(\cos x)^{-4/x^2}=(1-\sin^2x)^{-2/x^2}=\left(\left(1-{1\over\csc^2x} \right)^{\csc^2x}\right)^{-2\left(\sin x\over x\right)^2}$$

Can you (and/or your friend) take it from there?

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  • $\begingroup$ Yes! Thank you very much! $\endgroup$ – Alexandra Nov 13 '18 at 21:18

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