lim of function combining ln and trigonometry

is there a method (arithmetic simplification or something) which with i can solve lim that combine ln, e, and trigonometry? for example:$\displaystyle lim_{x\rightarrow0}\frac{ln(cosx)}{cosx-1} $ and:

$\displaystyle lim_{x\rightarrow\infty}\frac{2x-sin2x}{x^{2}+cos^{^{2}}x} $

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Re: lim of function combining ln and trigonometry

For the first one, simply use L'Hôpital's rule.

I attached a Wolfram Alpha solution for the second one in a pdf.

Re: lim of function combining ln and trigonometry

sin(2x) and $\displaystyle cos^2(x)$ are never larger than 1 so for large x, the fraction will be very close to $\displaystyle \frac{2x}{x^2}= \frac{2}{x}$ which goes to 0 as x goes to infinity.

Re: lim of function combining ln and trigonometry

Quote:

Originally Posted by

**orir** is there a method (arithmetic simplification or something) which with i can solve lim that combine ln, e, and trigonometry? for example:$\displaystyle \lim_{x\rightarrow0}\frac{ln(cosx)}{cosx-1} $ and:

$\displaystyle \lim_{x\rightarrow\infty}\frac{2x-sin2x}{x^{2}+cos^{^{2}}x} $

Without using L'Hopital, you can see $\displaystyle \frac{2-\frac{\sin(2x)}{x}}{x+\frac{cos^{^{2}}{x}}x} $.

Because both $\displaystyle \sin(2x)~\&~\cos^2(x)$ are bounded functions, the limit is clearly $\displaystyle 0~.$