# Find the limit of a rational function with a power

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 $$0$$.

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

• Please incorporate the question into the post instead of posting a link. Links tend to rot and many people don't like clicking through. Then show your work because it is much easier to give a good answer to what went wrong when we see it. – Ross Millikan Oct 16 '18 at 2:05
• Thank you. I will keep that in mind. I took a picture because I was having trouble writing my working out procedure as I am new to Latex. – David MB Oct 16 '18 at 2:53

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?
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 $$m=n^3$$.