Evaluate: $\lim_{n\to \infty}\left(\frac{a+n}{b+n}\right)^n$ I tried this way : $$\lim_{n\to \infty}\left(\frac{a+n}{b+n}\right)^n=\lim_{n\to \infty}\left(1+\frac{a-b}{b+n}\right)^n$$
We know $\lim_{n\to \infty}\left(1+\frac1u\right)^u=e$. therefor in this case we have: $$\lim_{n\to \infty}\left(1+\frac{a-b}{b+n}\right)^{\tfrac{b+n}{a-b}}=e$$
Here $n$ goes to infinity so we can ignore the number ($b$)added to it in numerator of the exponent. So $\lim_{n\to \infty}\left(1+\frac{a-b}{b+n}\right)^n=e^{a-b}$
Is my answer right? and is there any other approach to evaluate the limit?

I think it is easier to write the expression as $$\frac{(1+\frac{a}{n})^{n}}{(1+\frac{b}{n})^{n}}$$ and take it from there.
