Evaluate the limit-limx- 02 sin x- - sin 2 x-x3

We have, lim_x→ 0[2 sin x^∘ sin (2 x^∘)x^3] [ ∵ 1^∘ = π180 radians] = lim_x→ 0[2 sin(π x180) sin(2 π x180)x^3] [ ∵sin 2 θ = 2 sinθcosθ] = lim_x→ 0[2 si ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Substitution Method to Remove Indeterminate Form

Evaluate the ...

Question

Evaluate the limit:

$lim x \to 0 \frac{2 sin x^{\circ} - sin 2 x^{\circ}}{x^{3}}$

Open in App

Solution

We have,

$lim x \to 0 [\frac{2 sin x^{\circ} - sin (2 x^{\circ})}{x^{3}}]$

$[∵ 1^{\circ} = \frac{π}{180} r a d i a n s]$

$= lim x \to 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{2 sin (\frac{π x}{180}) - sin (\frac{2 π x}{180})}{x^{3}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$[∵ sin 2 θ = 2 sin θ cos θ]$

$= lim x \to 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{2 sin (\frac{π x}{180}) - 2 sin (\frac{π x}{180}) \times c o s (\frac{π x}{180})}{x^{3}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$= lim x \to 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{2 sin (\frac{π x}{180}) [1 - c o s (\frac{π x}{180})]}{x^{3}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$= lim x \to 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{2 sin (\frac{π x}{180}) \times 2 {sin}^{2} (\frac{π x}{360})}{x \times x^{2}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$[∵ 1 - cos θ = 2 {sin}^{2} \frac{θ}{2}]$

$lim x \to 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{4 sin [\frac{π x}{180}] \times s i n^{2} [\frac{π x}{360}]}{\frac{π x}{180} \times \frac{π x}{360} \times \frac{π x}{360}} \times \frac{π}{180} \times {(\frac{π}{360})}^{2} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$= 4 lim x \to 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \frac{s i n (\frac{π x}{180})}{\frac{π x}{180}} \times \frac{sin (\frac{π x}{360})}{\frac{π x}{360}} \times \frac{s i n (\frac{π x}{360})}{\frac{π x}{360}} \times \frac{π^{3}}{180 \times 360^{2}} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦$

$= 4 \times 1 \times 1 \times 1 \times \frac{π^{3}}{180 \times 360 \times 360}$

$[∵ lim θ \to 0 (\frac{sin θ}{θ}) = 1, θ i s i n r a d i a n s]$

$= {(\frac{π}{180})}^{3}$

Suggest Corrections

Evaluate the limit: limx→02sinx∘−sin2x∘x3

Evaluate the limit:

$lim x \to 0 \frac{2 sin x^{\circ} - sin 2 x^{\circ}}{x^{3}}$