Evaluating a limit I

Question

Consider the limit \begin{align} \lim_{x \to \infty} \left[ \frac{(x+a)^{x+1}}{(x+b)^{x}} - \frac{(x+a-n)^{x+1-n}}{(x+b-n)^{x-n}} \right]. \end{align} It is speculated that the resulting value is exponential in nature. What are some processes to demonstrate this speculation?

Answer 1

Let's find asymptotics of $$ f(x)=\frac{(x+a)^{x+1}}{(x+b)^x}. $$ Your limit is $$ \lim_{x\to\infty} (f(x) - f(x-n)). $$ So, $$ f(x)=\frac{(x+a)^{x+1}}{(x+b)^x} = (x+a) \frac{x^x \left(1+\frac ax\right)^x}{x^x \left(1+\frac bx\right)^x} = (x+a)\left(e^{a-b} + \frac12e^{a-b}(b^2-a^2)\frac1x + o\left(\frac1x\right)\right) $$ and your limit is $$ \lim_{x\to\infty} (f(x) - f(x-n)) = ne^{a-b}. $$