# lim of function combining ln and trigonometry

• June 13th 2013, 06:28 AM
orir
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: $lim_{x\rightarrow0}\frac{ln(cosx)}{cosx-1}$ and:
$lim_{x\rightarrow\infty}\frac{2x-sin2x}{x^{2}+cos^{^{2}}x}$
• June 13th 2013, 07:00 AM
ReneG
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.
• June 13th 2013, 07:43 AM
HallsofIvy
Re: lim of function combining ln and trigonometry
sin(2x) and $cos^2(x)$ are never larger than 1 so for large x, the fraction will be very close to $\frac{2x}{x^2}= \frac{2}{x}$ which goes to 0 as x goes to infinity.
• June 13th 2013, 07:43 AM
Plato
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: $\lim_{x\rightarrow0}\frac{ln(cosx)}{cosx-1}$ and:
$\lim_{x\rightarrow\infty}\frac{2x-sin2x}{x^{2}+cos^{^{2}}x}$

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

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