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lim-x-pi-20-1-tan-5x-sin-5x-cos-5x-




Question Number 111907 by bemath last updated on 05/Sep/20
lim_(x→(π/(20)))  ((1−tan 5x)/(sin 5x−cos 5x)) ?
$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{20}}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{tan}\:\mathrm{5}{x}}{\mathrm{sin}\:\mathrm{5}{x}−\mathrm{cos}\:\mathrm{5}{x}}\:? \\ $$
Answered by john santu last updated on 05/Sep/20
Answered by john santu last updated on 05/Sep/20
another way   lim_(x→(π/(20)))  ((cos 5x−sin 5x)/(sin 5x−cos 5x)) . (1/(cos 5x))  = −lim_(x→(π/(20)))  (1/(cos 5x)) = −(1/(cos ((π/4))))=−(√2)
$${another}\:{way}\: \\ $$$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{20}}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{5}{x}−\mathrm{sin}\:\mathrm{5}{x}}{\mathrm{sin}\:\mathrm{5}{x}−\mathrm{cos}\:\mathrm{5}{x}}\:.\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{5}{x}} \\ $$$$=\:−\underset{{x}\rightarrow\frac{\pi}{\mathrm{20}}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{cos}\:\mathrm{5}{x}}\:=\:−\frac{\mathrm{1}}{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{4}}\right)}=−\sqrt{\mathrm{2}} \\ $$

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