Lim x -3-1-cos 6 x-23-5-x

Lim_x →π/3√(1 cos 6x)/2(π/3 x) =lim_x →π/3√(2 sin^2 3x)/2(π/3 x) [∵ 1 cos θ=2 sin^2 θ] =lim_x →π/3√(2)sin 3x/√(2)(π/3 x) =lim_x →π/3sin 3x/(π/3 x) =lim_h → 0sin ...

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

Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x →π/3√1-...

Question

${lim}_{x \to \frac{π}{3}} \frac{\sqrt{1 - c o s 6 x}}{2 (\frac{π}{3} - x)}$

Open in App

Solution

${lim}_{x \to \frac{π}{3}} \frac{\sqrt{1 - c o s 6 x}}{2 (\frac{π}{3} - x)}$

$= {lim}_{x \to \frac{π}{3}} \frac{\sqrt{2 s i n^{2} 3 x}}{2 (\frac{π}{3} - x)}$

$[∵ 1 - c o s θ = 2 s i n^{2} θ]$

$= {lim}_{x \to \frac{π}{3}} \frac{\sqrt{2} s i n 3 x}{\sqrt{2} (\frac{π}{3} - x)}$

$= {lim}_{x \to \frac{π}{3}} \frac{s i n 3 x}{(\frac{π}{3} - x)}$

$= {lim}_{h \to 0} \frac{s i n 3 (\frac{π}{3} + h)}{(\frac{π}{3} - (\frac{π}{3} + h))}$ [Put $x = \frac{π}{3} + h]$

$= {lim}_{h \to 0} \frac{s i n (π + 3 h)}{- h}$

$= {lim}_{h \to 0} \frac{- s i n 3 h}{- h} [S i n (π + θ) = - s i n θ]$

$= 3 \times {lim}_{h \to 0} \frac{s i n 3 h}{h}$

$= 3 \times 1 [{lim}_{θ \to 0} \frac{s i n θ}{θ} = 1]$

=3

Suggest Corrections

Similar questions

Evaluate the following one sided limits:

$(i) {lim}_{x \to 2^{+}} \frac{x - 3}{x^{2} - 4}$

$(i i) {lim}_{x \to 2^{-}} \frac{x - 3}{x^{2} - 4}$

$(i i i) {lim}_{x \to 0^{+}} \frac{1}{3 x}$

$(i v) {lim}_{x \to 8^{+}} \frac{2 x}{x + 8}$

$(v) {lim}_{x \to 0^{+}} \frac{2}{x^{\frac{1}{5}}}$

$(v i) {lim}_{x \to \frac{π^{-}}{2}} t a n x$

$(v i i) {lim}_{x \to \frac{π}{2^{+}}} s e c x$

$(v i i i) {lim}_{x \to 0^{-}} \frac{x^{2} - 3 x + 2}{x^{3} - 2 x^{2}}$

$(i x) {lim}_{x \to - 2^{+}} \frac{x^{2} - 1}{2 x + 4}$

$(x) {lim}_{x \to 0^{+}} (2 - c o t x)$

$(x i) {lim}_{x \to 0^{-}} 1 + c o s e c x$

Evaluate:

lim x \to \frac{π}{3} \frac{sin (\frac{π}{3} - x)}{2 cos x - 1}

\lim_{x \to 0} \frac{1 - \cos 5 x}{1 - \cos 6 x}

Q. Solve:

lim x \to π / 6 \frac{2 sin x - 1}{4 {cos}^{2} x - 3}

lim x \to \frac{π}{3} \frac{sin (\frac{π}{3} - x)}{2 cos x - 1}

is equal to

Join BYJU'S Learning Program

limx→π3√1−cos6x2(π3−x)

${lim}_{x \to \frac{π}{3}} \frac{\sqrt{1 - c o s 6 x}}{2 (\frac{π}{3} - x)}$