i'm trying to find the following limit, if it exists, $$\lim_{n→ ∞} \frac{(n+7)^{n-5}}{n^n}$$.
Now, I've tried division like $$\lim_{n→ ∞}\frac{|n+1|}{|n|}$$, or dividing by the highest power, but I can't solve it.
Also thought of converting it to a function and using L'Hôpital's rule, but that seems more complicated than it should be. I think it's a simple limit to solve, but somehow I'm stuck and can't think of a way to tackle it.
Can someone give a hint or method; even a complete solution, but explaining the thought process?
Usually the method is to look for similariarity to patterns you've seen before and know the limits of. In this case, there is a limit $n\to \infty$ , there are exponents $n$ and a fraction and this immediately reminds me of the definition of Euler's constant $e$ and generally the term $e^x$.
The definition of $e$ I have in mind is
$$e = \lim_{n\to \infty}\left(1+\frac1n\right)^n$$
In order to get the limit at hand in that direction, we manipulate it to have a fraction term where both enumerator and denominator have exponent $n$, then take the exponent out of the fraction:
$$\frac{(n+7)^{n-5}}{n^n} = \frac{(n+7)^{n}}{n^n}\frac1{(n+7)^5} = \left(\frac{n+7}n\right)^n\frac1{(n+7)^5} = \left(1+\frac7n\right)^n\frac1{(n+7)^5}$$
This is just a manipulation of terms, but with a goal. They key insight is that if we now go to the limit
$$\lim_{n\to\infty}\frac{(n+7)^{n-5}}{n^n} = \lim_{n\to\infty}\left[\left(1+\frac7n\right)^n\frac1{(n+7)^5}\right],$$
the first term on the right hand side goes to $e^7$:
$$\lim_{n\to\infty}\left(1+\frac7n\right)^n = e^7$$
You can find this limit for example here: https://en.wikipedia.org/wiki/Exponential_function
The second part of the right hand side is much easier and elementary: $$\lim_{n\to\infty}\frac1{(n+7)^5} = 0.$$
So we finally can apply the theorem that the limit of a product is the product of the limits (if they exist and are not $\infty$):
$$\lim_{n\to\infty}\frac{(n+7)^{n-5}}{n^n} = \lim_{n\to\infty}\left[\left(1+\frac7n\right)^n\frac1{(n+7)^5}\right] = \lim_{n\to\infty}\left(1+\frac7n\right)^n \lim_{n\to\infty}\frac1{(n+7)^5} = e^7\times0 = 0$$