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  •  If    <img alt="\mathrm{f(x)=\left\{\begin{array}{cc} \frac{\log (1+a x)-\log (1-b x)}{x}, & x \neq 0 \\ k & , x=0 \end{array}\right.}" src="https://entrancecorner.oncodecog

 If    \mathrm{f(x)=\left\{\begin{array}{cc} \frac{\log (1+a x)-\log (1-b x)}{x}, & x \neq 0 \\ k & , x=0 \end{array}\right.}       and  f(x)  is continuous at  x=0  then the value of  k  is 

Option: 1

a - b


Option: 2

a + b


Option: 3

\mathrm{\log a+\log b }


Option: 4

none of these.


Answers (1)

best_answer

 Since f(x) is continuous at x=0, therefore

\mathrm{ \lim _{x \rightarrow 0} f(x)=f(0) \Rightarrow \lim _{x \rightarrow 0} f(x)=k \\ }

\mathrm{ \Rightarrow \lim _{x \rightarrow 0} \frac{\log (1+a x)-\log (1-b x)}{x}=k \\ }
\mathrm{\Rightarrow a \lim _{x \rightarrow 0} \frac{\log (1+a x)}{a x}-(-b) \lim _{x \rightarrow 0} \frac{\log (1-b x)}{-b x}=k \\ }
\mathrm{\Rightarrow a+b=k . }
 

Posted by

Pankaj

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