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Question Number 35727 by abdo mathsup 649 cc last updated on 22/May/18

$$find{lim}_{x\to 0}\frac{sin\left(shx\right)-sh\left(sinx\right)}{x}$$

Commented by abdo mathsup 649 cc last updated on 23/May/18

$$itsbetertousehospitaltheoreminthiscase$$$$letputf\left(x\right)=sin\left(shx\right)-sh\left(sinx\right)and$$$$g\left(x\right)=xwehave$$$$f^{{}^{\prime }}\left(x\right)=ch\left(x\right)cos\left(shx\right)-cos\left(x\right)ch\left(sinx\right)\Rightarrow$$$${lim}_{x\to 0}f\left(x\right)=1-1=0$$$$g^{{}^{\prime }}\left(x\right)=1so{lim}_{x\to 0}\frac{sin\left(shx\right)-sh\left(sinx\right)}{x}=0$$

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