Show that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$

Question

Could someone please show me the algebraic steps in showing that $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \frac{1}{2}$ as $x \rightarrow \infty$?

As the way I see it $\frac{1}{2}x^{\frac{2}{x}} \rightarrow \infty$ as $x \rightarrow \infty$

Thanks in advance.

Answer 1

$x^{\frac{2}{x}}=\exp\left(2\frac{\ln(x)}{x}\right)$, $\forall x>0$.

Then, use the fact that $\frac{\ln(x)}{x}$ goes to $0$ as $x$ goes to $+\infty$.

Answer 2

hint: Show $x^{\frac{1}{x}} \to 1$ as $x \to \infty$. To this end, we have: $\dfrac{\ln x}{x} \to 0$ by L'hopitale rule hence the result follows.

Answer 3

let y denote the expression given Then ln y= 1/2+(2/x)ln x. Then (2/x)ln x->0 as x->inf. Thus ln y ->1/2. Now take the exponential to get the answer.