Question Number 91157 by jagoll last updated on 28/Apr/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3tan}\:\mathrm{4}{x}−\mathrm{4tan}\:\mathrm{3}{x}}{\mathrm{3sin}\:\mathrm{4}{x}−\mathrm{4sin}\:\mathrm{3}{x}}\:=\:? \\ $$
Commented by john santu last updated on 28/Apr/20
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3}\left(\mathrm{4}{x}+\frac{\left(\mathrm{4}{x}\right)^{\mathrm{3}} }{\mathrm{3}}\right)−\mathrm{4}\left(\mathrm{3}{x}+\frac{\left(\mathrm{3}{x}\right)^{\mathrm{3}} }{\mathrm{3}}\right)}{\mathrm{3}\left(\mathrm{4}{x}−\frac{\left(\mathrm{4}{x}\right)^{\mathrm{3}} }{\mathrm{6}}\right)−\mathrm{4}\left(\mathrm{3}{x}−\frac{\left(\mathrm{3}{x}\right)^{\mathrm{3}} }{\mathrm{6}}\right)}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{12}{x}+\mathrm{64}{x}^{\mathrm{3}} \right)−\left(\mathrm{12}{x}+\mathrm{36}{x}^{\mathrm{3}} \right)}{\left(\mathrm{12}{x}−\mathrm{32}{x}^{\mathrm{3}} \right)−\left(\mathrm{12}{x}−\mathrm{18}{x}^{\mathrm{3}} \right)}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{28}{x}^{\mathrm{3}} }{−\mathrm{14}{x}^{\mathrm{3}} }\:=\:−\mathrm{2}\: \\ $$