<s>[INST] Convergence Proof: $\lim_{x\rightarrow\infty} \sqrt{4x+x^2}- \sqrt{x^2+x}$ I have to check whether the following expression converges; if yes I have to give the limit.
$$\lim_{x\rightarrow\infty} \sqrt{4x+x^2}- \sqrt{x^2+x}$$
Now I did the following:
$$\lim_{x\rightarrow\infty} \sqrt{4x+x^2}- \sqrt{x^2+x}$$
$$\lim_{x\rightarrow\infty} x\sqrt{\frac{4}{x}+1}- x\sqrt{1+\frac{1}{x}}$$
$$\lim_{x\rightarrow\infty} x \lim_{x\rightarrow\infty}(\sqrt{\frac{4}{x}+1}- \sqrt{1+\frac{1}{x}})$$
That confuses me. The left limit approaches $\infty$ while the right approaches $0$. What is wrong here or what can I conclude from that?
Thank you very muvh for your help in advance!
FunkyPeanut
 [/INST] You get nowhere by reasoning that way. The canonical way to work on this is "rationalizing": you multiply and divide by $\sqrt{4x+x^2}+\sqrt{x^2+x}$. That way you get
$$
\sqrt{4x+x^2}-\sqrt{x^2+x}=\frac{4x+x^2-x^2-x}{\sqrt{4x+x^2}+\sqrt{x^2+x}}=\frac{3x}{\sqrt{4x+x^2}+\sqrt{x^2+x}}=\frac3{\sqrt{4/x+1}+\sqrt{1+1/x}}\to\frac32
$$
</s>