# find the limit

• Oct 1st 2011, 04:53 PM
LAPOSH42
find the limit
lim ->∞ 1/x+1

x^{1/x} as x approaches infinity? I know that limit of (1+1/x)^{x} = e (as x-> infty), and limit of (1+x)^{1/x} = e (as x-> 0), but this is slightly different.

Any help is appreciated
• Oct 1st 2011, 05:03 PM
mr fantastic
Re: find the limit
Quote:

Originally Posted by LAPOSH42
lim ->∞ 1/x+1

x^{1/x} as x approaches infinity? I know that limit of (1+1/x)^{x} = e (as x-> infty), and limit of (1+x)^{1/x} = e (as x-> 0), but this is slightly different.

Any help is appreciated

Note that $\displaystyle x^{1/x} = e^{\frac{\ln(x)}{x}}$ so I suggest you apply l'Hospital's theorem to $\displaystyle \frac{\ln(x)}{x}$.
• Oct 1st 2011, 05:38 PM
Prove It
Re: find the limit
Quote:

Originally Posted by mr fantastic
Note that $\displaystyle x^{1/x} = e^{\frac{\ln(x)}{x}}$ so I suggest you apply l'Hospital's theorem to $\displaystyle \frac{\ln(x)}{x}$.

And also note that due to the continuity of the exponential function, that $\displaystyle \displaystyle \lim_{x \to a}e^{f(x)} = e^{\lim_{x \to a}f(x)}$