Question

Find the limit as x approaches infinity of `x ^(1 /x)`?

Answer 1

By using logarithmic properties,
`lim_{x to infty}x^{1/x}=1`

Let us look at some details.
Since `x=e^{lnx}`,
`lim_{x to infty}x^{1/x}=lim_{x to infty}e^{lnx^{1/x}}`
by the property `ln x^r=rlnx`,
`=lim_{x to infty}e^{{lnx}/x}`
by squeeze the limit in the exponent,
`=e^{lim_{x to infty}{lnx}/x}`
by l'Hopital's Rule,
`e^{lim_{x to infty}{1/x}/{1}}=e^0=1`