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Question Number 189757 by Shrinava last updated on 21/Mar/23

Find: lim_(xβ†’(𝛑/2)) (((√2) ∣ sin 4x ∣)/( (√(1 βˆ’ cos 4x)))) = ? Find: lim_(x→𝛑) (((√2) ∣ sin 6x ∣)/( (√(1 βˆ’ cos 6x)))) = ?

$$\mathrm{Find}:\:\:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\frac{\boldsymbol{\pi}}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}}\:\mid\:\mathrm{sin}\:\mathrm{4x}\:\mid}{\:\sqrt{\mathrm{1}\:βˆ’\:\mathrm{cos}\:\mathrm{4x}}}\:\:\:=\:\:\:? \\ $$$$\mathrm{Find}:\:\:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\boldsymbol{\pi}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}}\:\mid\:\mathrm{sin}\:\mathrm{6x}\:\mid}{\:\sqrt{\mathrm{1}\:βˆ’\:\mathrm{cos}\:\mathrm{6x}}}\:\:\:=\:\:\:? \\ $$

Answered by Frix last updated on 21/Mar/23

Use t=tan 2x in the 1^(st) and t=tan 3x in the 2^(nd) with t→0 ⇒ answer is 2

$$\mathrm{Use}\:{t}=\mathrm{tan}\:\mathrm{2}{x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{and}\:{t}=\mathrm{tan}\:\mathrm{3}{x}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{2}^{\mathrm{nd}} \:\mathrm{with}\:{t}\rightarrow\mathrm{0}\:\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\mathrm{2} \\ $$

Answered by mehdee42 last updated on 21/Mar/23

1)lim_(xβ†’(Ο€/2)) ((2(√2)∣sin2xcos2x∣)/( (√2)∣sin2x∣))=lim_(xβ†’(Ο€/2)) 2∣cos2x∣=2 2)lim_(xβ†’Ο€) ((2(√2)∣sin3xcos3x∣)/( (√2)∣sin3x∣))=lim_(xβ†’Ο€) 2∣cos3x∣=2

$$\left.\mathrm{1}\right){li}\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {{m}}\frac{\mathrm{2}\sqrt{\mathrm{2}}\mid{sin}\mathrm{2}{xcos}\mathrm{2}{x}\mid}{\:\sqrt{\mathrm{2}}\mid{sin}\mathrm{2}{x}\mid}={li}\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {{m}}\mathrm{2}\mid{cos}\mathrm{2}{x}\mid=\mathrm{2} \\ $$$$\left.\mathrm{2}\right){li}\underset{{x}\rightarrow\pi} {{m}}\frac{\mathrm{2}\sqrt{\mathrm{2}}\mid{sin}\mathrm{3}{xcos}\mathrm{3}{x}\mid}{\:\sqrt{\mathrm{2}}\mid{sin}\mathrm{3}{x}\mid}={li}\underset{{x}\rightarrow\pi} {{m}}\mathrm{2}\mid{cos}\mathrm{3}{x}\mid=\mathrm{2} \\ $$$$ \\ $$

Commented by Shrinava last updated on 24/Mar/23

thank you professor

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{professor} \\ $$

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