Question

HELP! limx−>-∞ of (√(x^2+4x)−√(x^2−5x))??

Answer 1

hmm you have to rationalise the function first..then all the things fall back into place

Answer 2

\[\lim_{x\rightarrow -\infty}\left(\sqrt{x^2+4x}-\sqrt{x^2-5x}\right)=\]\[=\lim_{x\rightarrow -\infty}\left(\sqrt{x^2+4x}-\sqrt{x^2-5x}\right)\frac{\sqrt{x^2+4x}+\sqrt{x^2-5x}}{\sqrt{x^2+4x}+\sqrt{x^2-5x}}=\]\[=\lim_{x\rightarrow -\infty}\frac{x^2+4x-(x^2-5x)}{\sqrt{x^2+4x}+\sqrt{x^2-5x}}=\lim_{x\rightarrow -\infty}\frac{9x}{|x|(\sqrt{1+4/x}+\sqrt{1-5/x})}=\]\[=\lim_{x\rightarrow -\infty}\frac{9x}{2|x|}=\lim_{x\rightarrow -\infty}\frac{9x}{-2x}=-\frac{9}{2}\]