How do you prove that the limit of x^(1/x) = 1 as x approaches infinity?

Question

Thanks in advance.

Answer 1

Let
y = x^(1/x)
ln y = ln[x^(1/x)] = ln x / x

Limit as x→∞ of {ln x / x} is of the indeterminant form ∞/∞

so we can apply L'Hospital's rule. The derivative of the quotient is the same as the quotient of the derivative of the numerator and the derivative of the denominator.

Limit as x→∞ of {ln x / x}
= Limit as x→∞ of {(1/x) / 1}
= Limit as x→∞ of 1/x = 0

ln y = 0
y = e^0 = 1

qed

Answer 2

if x approaches infinity the 1/x is 0 then if infinity is raised to power of zero,it is 1.

Answer 3

as x approaches infinity,

1/x will become 0

since any number with power of 0 will be 1, therefore

x^0 = 1

Answer 4

it is exactly as root says!