If limx→∞ x+cx−cx=4, then the value of c is

If $\underset{x\to \mathrm{\infty }}{lim} {\left(\frac{x+c}{x-c}\right)}^{x}=4,$ then the value of $c$ is

1. A

${\mathrm{log}}_{10}2$

2. B

${\mathrm{log}}_{c}2$

3. C

2

4. D

none of these

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Solution:

We have,

$\begin{array}{l}\underset{x\to \mathrm{\infty }}{lim} {\left(\frac{x+c}{x-c}\right)}^{x}=4\\ ⇒\underset{x\to \mathrm{\infty }}{lim} {\left(1+\frac{2c}{x-c}\right)}^{x}=4\\ ={e}^{\underset{x\to \mathrm{\infty }}{lim} \frac{2cx}{x-c}}=4⇒{e}^{2c}=4⇒{e}^{c}=2⇒c={\mathrm{log}}_{e}2\end{array}$

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