Rigorous method for showing this limit 
Prove the following limit; $$\lim_{x \to +\infty}\dfrac{\exp(x^2)}{10^{|x|}}$$  

The limit of this is $=\infty$
But what is the best method to show this:
L'Hospital doesn't seem very helpful here.
i.e as $x\to\infty$ the function becomes $\dfrac{\exp(x^2)}{10^{x}}$.
This is a $\infty/\infty$ type limit; now differentiating $$\dfrac{2x\exp(x^2)}{\ln(10)10^x}$$ this is again a $\infty / \infty$ type limit differentiating again;
$$\dfrac{4x^2\exp(x^2)+2\exp(x^2)}{\ln(10)^210^x}$$ again this is a $\infty/\infty$ limit. 
What is the best way to show that the limit is actually $+\infty$?

Hint: Note that $a^b=e^{b\ln a}$, so $10^{|x|}=e^{|x|\ln 10}$. Thus,
$$\lim_{x\to+\infty}\dfrac{e^{x^2}}{10^{|x|}}=\exp\left(\lim_{x\to+\infty}\left(x^2-|x|\ln 10\right)\right)$$
