[tex]\displaystyle\lim_{x\to\infty}\frac{2^x}{x^{10}}=\lim_{x\to\infty}\frac{e^{x\ln2}}{x^{10}}=\dfrac\infty\infty[/tex]
A few applications of L'Hopital's rule gives a decent idea of how this limit will ultimately behave.
[tex]=\displaystyle\lim_{x\to\infty}\frac{\ln2\,e^{x\ln2}}{10x^9}[/tex]
[tex]=\displaystyle\lim_{x\to\infty}\frac{(\ln2)^2\,e^{x\ln2}}{90x^8}[/tex]
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 [tex]\infty[/tex] as [tex]x\to\infty[/tex].
