# Find the limit as x approaches infinity of (x^(ln2))/(1+lnx)?

${\lim}_{x \to \infty} \frac{{x}^{\ln 2}}{1 + \ln x} = \infty$.
Intuitively, Imagine that this is a fight between the numerator and the denominator, and ${x}^{\ln 2}$ grows faster than $1 + \ln x$ as x approaches $\infty$, which means that the numerator will be larger than the denominator; therefore, the quotient tends to infinity.
lim_{x to infty}{x^{ln2}}/{1+lnx} =lim_{x to infty}{(ln2)x^{ln2-1}}/{1/x} = (ln2)lim_{x to infty}{x^{ln2}x^(-1)}/{x^-1}
$= \left(\ln 2\right) {\lim}_{x \to \infty} {x}^{\ln 2} = \infty$