If _ x ( x^2 + 5x ) - 2 ( cx + 1 ) = - 2 then - Vidyadip

MCQ

If $\mathop {\lim }\limits_{x \to \infty } \left\{ {\ln \left( {{x^2} + 5x} \right) - 2\ln \left( {cx + 1} \right)} \right\} = - 2$ then

✓
$c = e$
B
$c = e^{-1}$
C
$c=-e$
D
None of these

✓

Answer

Correct option: A.

$c = e$

a
$\mathop {\lim }\limits_{x \to \infty } \left\{ {\ln \left( {{x^2} + 5x} \right) - 2\ln \left( {cx + 1} \right)} \right\}$

$=\ln \left\{\frac{\left(x^{2}+5 x\right)}{(c x+1)^{2}}\right\}=-2$

$\frac{x^{2}+5 x}{c^{2} x^{2}+2 c x+1}=e^{-2}$

$\frac{1}{c^{2}}=\frac{1}{e^{2}} \Rightarrow c=e$

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