Question

Limit x approach infinity x[ ( 1 + a/x )^1+1/x - x^-1/x ( x+ a ) ]

Answer 1

$\lim \limits_{x\to \infty} x[ ( 1 + \frac{a}{x} )+ \frac{1}{x} - \frac{x^{-1}}{x} ( x+ a ) ]=$$\lim \limits_{x\to \infty} [ ( x + a )+ 1 - ( 1+ \frac{a}{x} ) ]=$ $\lim \limits_{x\to \infty} [ x + a - \frac{a}{x} ]=\infty$

As we know $\lim \limits_{x\to \infty} \frac{1}{x}=0$ and $\lim \limits_{x\to \infty} x=\infty$. so the expression increases indefinitely.