_ x [(2 x+3)^7(x-5)^3 (2 x-5)^ 10 ]=

MCQ

$\lim _{x \rightarrow \infty}\left[\frac{(2 x+3)^7(x-5)^3}{(2 x-5)^{10}}\right]=$

A
$\frac{3}{8}$
✓
$\frac{1}{8}$
C
$\frac{1}{6}$
D
$\frac{1}{4}$

✓

Answer

Correct option: B.

$\frac{1}{8}$

(B) $\frac{1}{8}$
Hint:
$\lim _{x \rightarrow \infty} \frac{(2 x+3)^7 \cdot(x-5)^3}{(2 x-5)^{10}} $
$=\frac{\lim _{x \rightarrow \infty}\left(\frac{2 x+3}{x}\right)^7 \cdot\left(\frac{x-5}{x}\right)^3}{\lim _{x \rightarrow \infty}\left(\frac{2 x-5}{x}\right)^{10}} $
$=\frac{\lim _{x \rightarrow \infty}\left(2+\frac{3}{x}\right)^7 \times \lim _{x \rightarrow x}\left(1-\frac{5}{x}\right)^3}{\lim _{x \rightarrow \infty}\left(2-\frac{5}{x}\right)^{10}}$
$=\frac{(2+0)^7 \times(1-0)^3}{(2-0)^{10}} \quad \ldots\left[\lim _{x \rightarrow \infty} \frac{1}{x^k}=0, \mathrm{k}>0\right]$
$=\frac{1}{8}$

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