-3
$\begingroup$

Do you have some idea how to calculate limit without using L'Hospital rule? $$\lim_{x\to0} \frac{2^x-\cos\left(x\right)}{3^x-\operatorname{ch}\left(x\right)}$$

Thank you for help?

$\endgroup$
  • 2
    $\begingroup$ What is "ch(x)"? Hyperbolic cosine? $\endgroup$ – DonAntonio Jan 9 '14 at 19:57
  • $\begingroup$ @DonAntonio Some country uses the notations $ch(x)$ and $sh(x)$ and $th(x)$ and $tg(x)$ and $cotg(x)$ for $\cosh(x)$ and $\sinh(x)$ and $\tanh(x)$ and $\cot(x)$ respectively. $\endgroup$ – user63181 Jan 9 '14 at 20:06
  • $\begingroup$ @SamiBenRomdhane Also, $ ctg (x)$ for $cotg(x)$. $\endgroup$ – TZakrevskiy Jan 9 '14 at 21:25
2
$\begingroup$

Using the Taylor series

$$\frac{2^x-\cos x}{3^x-\cosh x}\sim_0\frac{1+x\log2-1+o(x)}{1+x\log3-1+o(x)}\sim_0\frac{\log2}{\log3}$$

$\endgroup$
  • $\begingroup$ Nice using of Taylor expansion Sami+ $\endgroup$ – mrs Jan 11 '14 at 2:55
0
$\begingroup$

$$\frac{2^x-\cos x}{3^x-\cosh x}=\frac{\mathrm e^{x\log2}-\cos x}{\mathrm e^{x\log3}-\cos x}=\frac{1+x\log2+o(x)-(1+o(x))}{1+x\log3+o(x)-(1+o(x))}\to\frac{\log2}{\log3} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.