Question

Solve $\underset{x\to \infty }{lim}{\left(\frac{x+6}{x+1}\right)}^{x+4}$

Answer 1

Let, $A=\underset{x\to \infty }{lim}{\left(\frac{x+6}{x+1}\right)}^{x+4}=\underset{x\to \infty }{lim}{[1+\frac{5}{x+1}]}^{x+4}$
$=\underset{x\to \infty }{lim}{\left[{(1+\frac{5}{x+1})}^{\frac{x+1}{5}}\right]}^{\frac{5(x+4)}{x+1}}=e^5$
$[\because \underset{x\to \infty }{lim}{(1+\frac{5}{x+1})}^{\frac{x+1}{5}}=e\text{and}\underset{x\to \infty }{lim}\frac{5(x+4)}{x+1}=\underset{x\to \infty }{lim}\frac{5(1+4/x)}{1+1/x}=5].$