How to determine the limit of a quotient containing exponents involving n?

Question

I'm trying to work out the limit of the following:

$$\frac{n2^n}{(n+1)3^n + n^7}$$

but I'm having a hard time applying the algebra of limits to the $2^n$ and $3^n$ terms. Can anyone point me in the right direction please?

Thanks

Answer 1

HINT:

$$\frac{n\cdot 2^n}{(n+1)3^n+n^7}=\frac{\left(\frac23\right)^n}{\frac{n+1}n+\frac{n^7}{3^n}}$$

Now, we know if $\displaystyle|x|<1,\lim_{n\to\infty}x^n=0$

As $\displaystyle\lim_{n\to\infty}\frac{n^7}{3^n}$ is of the form $\frac\infty\infty$ we can apply L'Hospital's Rule repeatedly as long as the condition is satisfied to find the limit converges to $0$

Answer 2

We have $\frac{n(2^{n})}{(n+1)3^{n} + n^{7}}$ is always greater than $0$ for positive $n$. On the other hand, $\frac{n(2^{n})}{(n+1)3^{n} + n^{7}} \leq \frac{2^{n}}{3^{n}}$ which goes to $0$ as n goes to infinity. Thus, the limit should be $0$.