# lim of function combining ln and trigonometry

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

#### ReneG

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

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

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#### HallsofIvy

MHF Helper
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.

#### Plato

MHF Helper
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 $$\displaystyle 0~.$$