Question

$\lim_{x\to\infty} x^{\frac{1}{x}} = $

• A. 1
• B. $\infty$
• C. 0
• D. $none\, of \, these$

Answer 1

Answer: (A)

Let $ y = \lim_{x\to\infty} x^{\frac{1}{x}}$ .....(i)
Taking log in (i) on both sides, we get
$ \lim _{x\to \infty } \frac{1}{x } \log x $
Applying L'-Hospital's Rule, we get
$ \log y = \lim _{x\to \infty } \frac{\frac{1}{x}}{1} = \lim _{x\to \infty } \frac{1}{x} =0$
or $y = e^0 = 1$