Question

pls answer the question as soon as possible​

Answer 1

Answer:

• Option (A) $\bold{e^{12}}$

Explanation:

$\red \bigstar \: \tt lim_{x \rightarrow \infin}( \dfrac{x + 5}{x - 7} ) {}^{x} \\ \\ \leadsto \tt \: lim_{x \rightarrow \infin}( \frac{x(1 + \dfrac{5}{x}) }{x(1 - \dfrac{7}{x} )} ) {}^{x} \\ \\ \leadsto \tt \: lim_{x \rightarrow \infin}( \frac{ \cancel{x}(1 + \dfrac{5}{x}) }{ \cancel{x}(1 - \dfrac{7}{x} )} ) {}^{x} \\ \\ \tt \: We \: get \: \red{1 {}^{ \infty }} \: form, \\ \\ \leadsto \tt \: lim_{x \rightarrow \infin} \: e {}^{x} ( \dfrac{1 + \dfrac{5}{x} }{1 - \dfrac{7}{x} } - 1) \\ \\ \leadsto \tt \: lim_{x \rightarrow \infin} \: e {}^{x} ( \dfrac{1 + \dfrac{5}{x} - 1 + \dfrac{7}{x} }{1 - \dfrac{7}{x} } ) \\ \\ \leadsto \tt \: lim_{x \rightarrow \infin} \: e {}^{x}( \dfrac{ \dfrac{12}{ \cancel{x}} }{ \dfrac{x - 7}{ \cancel{x}} } ) \\ \\ \leadsto \tt \: lim_{x \rightarrow \infin} \: e {}^{x} ( \frac{12}{x - 7} ) \\ \\ \leadsto \tt \: lim_{x \rightarrow \infin} \: e {}^{( \frac{12}{ \frac{x}{x} - \frac{7}{x}} ) } \\ \\ \leadsto \tt \: lim_{x \rightarrow \infin} \: e {}^{ ( \frac{12}{ 1 - \frac{7} {x}} )} \\ \\ \leadsto \tt \: e {}^{ ( \dfrac{12}{1 - 0} ) }\\ \\ \leadsto \tt \: e {}^{12} \: \green \bigstar$

[ Note :- Refer to the attachment for important limits formulae.]