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  • Evaluate  <img alt="\mathrm{\lim _{x \rightarrow 0} \frac{\sinh (x)-\sin (x)}{x(\cosh (x)-\cos (x))}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bx%20%5Crigh

Evaluate  \mathrm{\lim _{x \rightarrow 0} \frac{\sinh (x)-\sin (x)}{x(\cosh (x)-\cos (x))}}

Option: 1

\frac{1}{3}


Option: 2

\frac{2}{5}


Option: 3

\frac{7}{5}


Option: 4

0


Answers (1)

best_answer

\mathrm{\lim _{x \rightarrow 0} \frac{\sinh (x)-\sin (x)}{x(\cosh (x)-\cos (x))} =\lim _{x \rightarrow 0} \frac{\left(x+\frac{x^{3}}{6}+o\left(x^{4}\right)\right)-\left(x-\frac{x^{3}}{6}+o\left(x^{4}\right)\right)}{x\left(\left(1+\frac{x^{2}}{2}+o\left(x^{3}\right)\right)-\left(1-\frac{x^{2}}{2}+o\left(x^{3}\right)\right)\right)} }
                                                     \mathrm{=\lim _{x \rightarrow 0} \frac{\frac{x^{3}}{3}+o\left(x^{4}\right)}{x+o\left(x^{4}\right)} }
                                                     \mathrm{=\lim _{x \rightarrow 0} \frac{\frac{1}{3}+o\left(x^{3}\right)}{1+o\left(x^{3}\right)} }
                                                     \mathrm{ =\frac{1}{3} }.


 

Posted by

Divya Prakash Singh

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