David Young

2021-12-30

Prove that:

$\underset{x\to 0}{lim}\frac{\mathrm{ln}\left(\mathrm{cos}x\right)}{\mathrm{ln}(1-\frac{{x}^{2}}{2})}=1$

without LHopitals rule.

without LHopitals rule.

Esta Hurtado

Beginner2021-12-31Added 39 answers

Using

Therefore

Gerald Lopez

Beginner2022-01-01Added 29 answers

Vasquez

Expert2022-01-09Added 669 answers

On OP's request I am converting my comment into an answer. The limit in question is easily solved if one rewrites the given expression as

And then the limit is easily seen to be

Other limits in your question (after the fold) don't have any problem as we can just plug x=0 to evaluate them (the functions concerned are continuous at 0).

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

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