# Solve the following limit

$$\lim_{n\rightarrow \infty} \frac{\sqrt{n+a}-\sqrt{n+b}}{\sqrt{n+c}-\sqrt{n+d}},\textrm{ } (c\neq d)$$ I really don't know what to use to solve this. Any help would be greatly appreciated.

Following your sugestions I get

$$\lim_{n\rightarrow \infty} \frac{\sqrt{n+a}-\sqrt{n+b}}{\sqrt{n+c}-\sqrt{n+d}}=\lim_{n\rightarrow \infty} \frac{\sqrt{n+a}+\sqrt{n+b}}{\sqrt{n+c}+\sqrt{n+d}}\left(\frac{a-b}{c-d}\right)=\left(\frac{a-b}{c-d}\right)$$

That should be it, thanks.

• Hint: Multiply numerator and denominator with $\sqrt{n+c}+\sqrt{n+d}$ – gammatester Nov 17 '16 at 15:16
• Another idea is to try binomial expansion for a few terms. – Simply Beautiful Art Nov 17 '16 at 15:18
• Also, multiply and divide by $\sqrt{n+a}+\sqrt{n+b}$. – Galc127 Nov 17 '16 at 15:18
• @Galc127 I presume you mean do this in addition to what gammatester suggests - carefully done that helps a lot. – Mark Bennet Nov 17 '16 at 15:27
• @MarkBennet, of course. We need to do both in order to evaluate the limit. – Galc127 Nov 17 '16 at 15:28

After factoring out by $\sqrt{n}$ and simplifying,
$$\sqrt{1+\frac{a}{n}}=1+\frac{a}{2n}(1+\epsilon(n)),$$
$$\frac{a-b}{c-d}$$