Find the limit of this:
lim n-infinity of (log(n+1) + a^(n+1)) / (log(n) + a^n)

Question

Find the limit of this:
lim n->infinity of (log(n+1) + a^(n+1)) / (log(n) + a^n)

anonymous · Answer

after doing l'hopital's rule once, i got
$$\lim_{n \rightarrow \infty} \frac{ 1/(n+1) + (n+1)a^n }{ 1/n + n * a^{n-1} }$$
any idea how to proceed after this?

anonymous · Answer

@tkhunny

tkhunny · Answer

A little algebra.

$\dfrac{n\cdot\left(1+(n+1)^2\cdot a^{n}\right)}{(n+1)\cdot\left(1+n^2\cdot a^{n+1}\right)}$

What is the value of 'a'?  Any restrictions?

You do know that l'Hospital doesn't always work, right?  I'm not saying it doesn't here.  Just mentioning it.

anonymous · Answer

Notice that the $\dfrac{1}{n+1}$ and $\dfrac{1}{n}$ terms disappear as $n\to\infty$. Meanwhile,
$$\frac{(n+1)a^n}{na^{n-1}}=a\frac{n+1}{n}$$