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Math Help - lim of function combining ln and trigonometry

  1. #1
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    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}
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  2. #2
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    Re: lim of function combining ln and trigonometry

    For the first one, simply use L'H˘pital's rule.

    I attached a Wolfram Alpha solution for the second one in a pdf.
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    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.
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    Re: lim of function combining ln and trigonometry

    Quote Originally Posted by orir View Post
    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~.
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