lim of function combining ln and trigonometry

Mar 2013
70
0
israel
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:
\(\displaystyle lim_{x\rightarrow\infty}\frac{2x-sin2x}{x^{2}+cos^{^{2}}x} \)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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
Aug 2006
22,492
8,653
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 \(\displaystyle 0~.\)