Calculus of a limit [Original question + solution]

Ah, ok, i just really try to avoid creating new posts to avoid spam and to keep things organized, i'll know from now on.

Thank you so much for your answer it actually helped me a ton. Once you know what you're doing it becomes so easy. At least i think i've done it correctly. :p

So my original problem was: $$\displaystyle \lim_{n\rightarrow \infty}\frac{5^{n-2}-3^{n+1}}{4^{n-1}+5^n}$$

The steps that i took: (i've skipped some steps as i cba to write every single detail, it would just take too much time)$$\displaystyle\lim_{n\rightarrow \infty}\frac{\frac{5^{n-2}}{5^n}}{\frac{4^{n-1}}{5^n}+\frac{5^n}{5^n}} - \lim_{n\rightarrow \infty}\frac{\frac{3^{n+1}}{5^n}}{\frac{4^{n-1}}{5^n}+\frac{5^n}{5^n}} = \lim_{n\rightarrow \infty}\frac{5^{-2}}{\left(\frac{4^n}{5^n}\right)*4^{-1} + 1} - \lim_{n\rightarrow \infty}\frac{\left(\frac{3^n}{5^n}\right)*3}{\left(\frac{4^n}{5^n}\right)*4^{-1} + 1} = \frac{\frac{1}{25}}{0+1} - \frac{0}{0+1} = \frac{1}{25}$$