(C) We know that $\lim _{x \rightarrow \infty} (1 + \frac{a}{x})^x = e^a$.Given expression: $\lim _{x \rightarrow \infty} (\frac{x+5}{x+2})^{x+3}$$= \

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(C) We know that $\lim _{x \rightarrow \infty} (1 + \frac{a}{x})^x = e^a$.Given expression: $\lim _{x \rightarrow \infty} (\frac{x+5}{x+2})^{x+3}$$= \lim _{x \rightarrow \infty} (1 + \frac{3}{x+2})^{x+3}$$= \lim _{x \rightarrow \infty} [(1 + \frac{3}{x+2})^{\frac{x+2}{3}}]^{\frac{3(x+3)}{x+2}}$$= e^{\lim _{x \rightarrow \infty} \frac{3x+9}{x+2}}$$= e^{\lim _{x \rightarrow \infty} \frac{3 + 9/x}{1 + 2/x}}$$= e^3$

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$\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}$ equals

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A
$e$
B
$e^2$
C
$e^3$
D
$e^5$

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$(a)$ Clostridium butylicum	$(i)$ Cyclosporin-$A$
$(b)$ Trichoderma polysporum	$(ii)$ Butyric acid
$(c)$ Monascus purpureus	$(iii)$ Citric acid
$(d)$ Aspergillus niger	$(iv)$ Blood cholesterol lowering agent

$\lim _{x \rightarrow \infty}\left(\frac{x+5}{x+2}\right)^{x+3}$ equals

Similar Questions

Assuming complete ionisation,the $pH$ of $0.1 \ M \ HCl$ is $1$. The molarity of $H_2SO_4$ with the same $pH$ is (in $M$)

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Match the following columns and select the correct option: $(a)$ Clostridium butylicum $(i)$ Cyclosporin-$A$ $(b)$ Trichoderma polysporum $(ii)$ Butyric acid $(c)$ Monascus purpureus $(iii)$ Citric acid $(d)$ Aspergillus niger $(iv)$ Blood cholesterol lowering agent $(a)\quad (b)\quad (c)\quad (d)$

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