I have a problem to find answer for this limit.limx→01−cos13x1−cos⁡(x13)

We need two standard limitslimx→01-cosxx2=12,limx→axn-anx-a=nan-1We can proceed as follows=limx→01−cos13x1−cos⁡x13=limx→01−cos13x1−cos⁡x⋅1−cos⁡x1−cos⁡x13=limt→11−t131−t⋅limx→01−cos⁡x1−cos⁡x13&nbsp;&nbsp;(putting&nbsp;&nbsp;t=cos⁡x)=13⋅1−23⋅limx→01−cos⁡xx2⋅x21−cos⁡x13=16⋅limx→0x43⋅x231−cos⁡x13=16⋅0⋅2=0

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burubukuamaw

Answered question

2022-03-31

I have a problem to find answer for this limit.

lim_{x \to 0} \frac{1 - \cos^{\frac{1}{3}} x}{1 - \cos (x^{\frac{1}{3}})}

Answer & Explanation

Ben Castillo

Beginner2022-04-01Added 13 answers

We need two standard limits
$\lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = \frac{1}{2}, \lim_{x \to a} \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}$
We can proceed as follows
$= lim_{x \to 0} \frac{1 - \cos^{\frac{1}{3}} x}{1 - {\cos x}^{\frac{1}{3}}}$
$= lim_{x \to 0} \frac{1 - \cos^{\frac{1}{3}} x}{1 - \cos x} \cdot \frac{1 - \cos x}{1 - {\cos x}^{\frac{1}{3}}}$
$= lim_{t \to 1} \frac{1 - t^{\frac{1}{3}}}{1 - t} \cdot lim_{x \to 0} \frac{1 - \cos x}{1 - {\cos x}^{\frac{1}{3}}} (putting t = \cos x)$
$= \frac{1}{3} \cdot 1^{- \frac{2}{3}} \cdot lim_{x \to 0} \frac{1 - \cos x}{x^{2}} \cdot \frac{x^{2}}{1 - {\cos x}^{\frac{1}{3}}}$
$= \frac{1}{6} \cdot lim_{x \to 0} x^{\frac{4}{3}} \cdot \frac{x^{\frac{2}{3}}}{1 - {\cos x}^{\frac{1}{3}}}$
$= \frac{1}{6} \cdot 0 \cdot 2 = 0$

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