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Q. $\displaystyle\lim _{x \rightarrow \infty} \frac{(2 x+1)^{40}(4 x-1)^{5}}{(2 x+3)^{45}}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow \infty} \frac{(2 x+1)^{40}(4 x-1)^{5}}{(2 x+3)^{45}}$
$=\displaystyle\lim _{x \rightarrow \infty} \frac{\left(2+\frac{1}{x}\right)^{40}\left(4-\frac{1}{x}\right)^{5}}{\left(2+\frac{3}{x}\right)^{45}}$
(Dividing numerator and denominator by $x^{45}$ )
$=\frac{2^{40} 4^{5}}{2^{45}} $
$=2^{5}=32$