# lim-x-4-2x-2-16-4-4x-2-16x-1-2x-2-16-4-

Question Number 131388 by Eric002 last updated on 04/Feb/21
$$\underset{{x}\rightarrow\mathrm{4}} {{lim}}\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}+\mathrm{4}}{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{16}{x}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}−\mathrm{4}} \\$$
Answered by bemath last updated on 04/Feb/21
$$\:\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\left(\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}+\mathrm{4}}{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{16}{x}}\:−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}−\mathrm{4}}\right)=? \\$$$$\left(\bullet\right)\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}\:−\mathrm{4}}\:=\:\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}\:+\mathrm{4}}{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{32}} \\$$$$\left(\bullet\bullet\right)\:\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}\:+\mathrm{4}}{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{16}{x}}−\frac{\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}+\mathrm{4}}{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{32}} \\$$$$\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\left(\sqrt{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{16}}+\mathrm{4}\right)×\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\mathrm{4}{x}^{\mathrm{2}} −\mathrm{16}{x}}−\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{32}}\right) \\$$$$=\:\mathrm{8}\:×\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{4}{x}\left({x}−\mathrm{4}\right)}−\frac{\mathrm{1}}{\mathrm{2}\left({x}−\mathrm{4}\right)\left({x}+\mathrm{4}\right)}\right) \\$$$$=\:\mathrm{8}×\:\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\left(\frac{{x}+\mathrm{4}}{\mathrm{4}{x}\left({x}−\mathrm{4}\right)\left({x}+\mathrm{4}\right)}−\frac{\mathrm{2}{x}}{\mathrm{4}{x}\left({x}−\mathrm{4}\right)\left({x}+\mathrm{4}\right)}\right) \\$$$$=\:\mathrm{8}×\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\left(\frac{−\left({x}−\mathrm{4}\right)}{\mathrm{4}{x}\left({x}−\mathrm{4}\right)\left({x}+\mathrm{4}\right)}\right) \\$$$$=\:\mathrm{8}×\:\underset{{x}\rightarrow\mathrm{4}} {\mathrm{lim}}\left(\frac{−\mathrm{1}}{\mathrm{4}{x}\left({x}+\mathrm{4}\right)}\right)=−\frac{\mathrm{8}}{\mathrm{16}×\mathrm{8}} \\$$$$=−\frac{\mathrm{1}}{\mathrm{16}} \\$$$$\:\:\:\:\:\: \\$$
Commented by bramlexs22 last updated on 04/Feb/21
$$\\$$$$\mathrm{Wow}\:\mathrm{es}\:\mathrm{ist}\:\mathrm{so}\:\mathrm{cool}\:\mathrm{Sir} \\$$
Commented by Eric002 last updated on 04/Feb/21
$${well}\:{done}\:{sir} \\$$