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  • Find value of given limit: <img alt="\mathrm{ \lim _{x \rightarrow 1} \frac{\cos y \cdot \log x}{x-1} }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20%5Clim%20_%7Bx

Find value of given limit: \mathrm{ \lim _{x \rightarrow 1} \frac{\cos y \cdot \log x}{x-1} }, where cos y is a constant.

Option: 1

\mathrm{\cos y}

 


Option: 2

\mathrm{\sin y}


Option: 3

1


Option: 4

0


Answers (1)

best_answer

\mathrm{ \lim _{x \rightarrow 1} \frac{\cos y \cdot \log x}{x-1} }

\mathrm{Let \, f(x)=\cos y \cdot \frac{\log x}{x-1}}

\mathrm{Let \: \mathrm{x}=1+\mathrm{h}\: , then \: \mathrm{x}\: tends\: to\: 1 \: when \: \mathrm{h}\: tends\: to \: 0 , then: }

\mathrm{\begin{aligned} & \lim _{x \rightarrow 1^{+}} f(x)=\lim _{h \rightarrow 0} f(1+h)=\lim _{h \rightarrow 0} \frac{\log (1+h)}{1+h-1} \\ & =\lim _{h \rightarrow 0} \frac{\operatorname{cosy}}{h} \log (1+h) \end{aligned} }

We know about the logarithmic result \mathrm{ \lim _{x \rightarrow 0} \frac{\log (1+x)}{x}=1}

Therefore, putting \mathrm{\mathrm{x}=\mathrm{h},} we get:

\mathrm{ \lim _{x \rightarrow 1} \frac{\cos y \cdot \log x}{x-1}=\cos y }
 

Posted by

Pankaj

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