Let the function f(x)= cc _ e (1+5 x)- _ e (1+ x) x & if x 0 10 &… - Vidyadip

MCQ

Let the function $f(x)=\left\{\begin{array}{cc}\frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & \text { if } x \neq 0 \\ 10 & \text {; if } x=0\end{array}\right.$ be continuous at $x=0$.The $\alpha$ is equal to.

A
$10$
B
$-10$
C
$5$
✓
$-5$

✓

Answer

Correct option: D.

$-5$

d
$f(x)=\left\{\begin{array}{cc}\frac{\ln (1+5 x)-\ln (1+\alpha x)}{x} & ; x \neq 0 \\ 10 & ; x=0\end{array}\right.$

$\lim _{x \rightarrow 0} \frac{\ln (1+5 x)-\ln (1+\alpha x)}{x}=10$

Using expension

$\lim _{x \rightarrow 0} \frac{(5 x+\ldots \ldots)-(\alpha x+\ldots \ldots)}{x}=10$

$5-\alpha=10 \Rightarrow \alpha=-5$

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