Evaluating limit $\lim_{n\to\infty}({\sqrt{4^n + 3^n} - 2^n})$

Question

I have to find: $$\lim_{n\to\infty}\left({\sqrt{4^n + 3^n} - 2^n}\right)$$

I plugged in some numbers and it seems as if this sequence were approaching infinity, but I do not know how to begin evaluating this.

If the exponent weren't so arbitrary, it would be possible to bring the radical expression to the denominator, but now I am stuck.

Answer 1

Hint: multiply by

$$\frac{\sqrt{4^n+3^n}+2^n}{\sqrt{4^n+3^n}+2^n}$$

Then divide both numerator and denominator by $3^n$. Keep in mind that $3^n=\sqrt{3^{2n}}=\sqrt{9^n}$.

Answer 2

Rewrite the expression as $$ 2^n \sqrt{1 + (3/4)^n} - 2^n = 2^n\left(\sqrt{1 + (3/4)^n} - 1\right) = \frac{\sqrt{1 + \epsilon^n} - 1}{\delta^{n}} $$
with $\epsilon = 3/4, \, \delta = 1/2$. Now use L'Hopital's Theorem. Since $\epsilon > \delta$, you will find that the limit is $\infty$.