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Question Number 97238 by bobhans last updated on 07/Jun/20

Commented by john santu last updated on 07/Jun/20

lim_(x→0)  ((tan^2 x (cos^2 x−1))/((1+(x^2 /4)−1)(1−((sin^2 x)/2)−1))) =  lim_(x→0)  ((−x^4 )/(−(1/8)x^4 )) = 8

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:^{\mathrm{2}} {x}\:\left(\mathrm{cos}\:^{\mathrm{2}} {x}−\mathrm{1}\right)}{\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{4}}−\mathrm{1}\right)\left(\mathrm{1}−\frac{\mathrm{sin}\:^{\mathrm{2}} {x}}{\mathrm{2}}−\mathrm{1}\right)}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{−{x}^{\mathrm{4}} }{−\frac{\mathrm{1}}{\mathrm{8}}{x}^{\mathrm{4}} }\:=\:\mathrm{8}\: \\ $$

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