Question

Lim x-->+ infinity (2^x)/x^10

Answer 1

$\displaystyle\lim_{x\to\infty}\frac{2^x}{x^{10}}=\lim_{x\to\infty}\frac{e^{x\ln2}}{x^{10}}=\dfrac\infty\infty$

A few applications of L'Hopital's rule gives a decent idea of how this limit will ultimately behave.

$=\displaystyle\lim_{x\to\infty}\frac{\ln2\,e^{x\ln2}}{10x^9}$
$=\displaystyle\lim_{x\to\infty}\frac{(\ln2)^2\,e^{x\ln2}}{90x^8}$

and so on. Notice that the numerator will consistently behave exponentially, while the denominator will eventually be rendered into a constant. This means the function diverges to $\infty$ as $x\to\infty$.