Find the limit of a rational function with a power

Question

I have been trying to find the limit of following question but can't seem to get the right answer. I first took the logs because the limit of the power is undefined and then tried to solve the limit using substitution. But I keep getting $1$, when the answer converges towards $0$.

$$\lim_{x\to\infty}\left(\frac{x^3}{x^3+1}\right)^{(3x^4+2)/x}$$

Answer 1

When you are taking limits as $x \to \infty$ you want terms like $\frac 1x$. This should prompt you to see $\frac {x^3}{x^3+1}=1-\frac 1{x^3+1}$

Now I will work informally-you need to justify this. The $+1$ doesn't matter in $\frac 1{x^3+1}$ so we are asking about $\left(1-\frac 1{x^3}\right)^{(3x^3+\frac 2x)}$ and the $\frac 2x$ doesn't matter so we have $\left(\left(1-\frac 1{x^3}\right)^{x^3}\right)^3$ Does the inside of the outer parentheses look familiar?

Answer 2

Hint: Disregard the +2 in $3n^4+2$ and simplify a bit. You end up with a lot of $n^3$ terms that are going to infinity. Try a substitution like $m=n^3$.