Lim x - 11-x-1-3-1-x-3

Lim_x → 1 (x^ 1/3 1)/ (x^ 2/3 1) =lim_x → 1x^ 1/3 1^ 1/3/x^ 2/3 1^ 2/3 =lim_x → 1x^ 1/3 1^ 1/3/x^ 2/3 1^ 2/3/x 1 [Dividing numerator and drominator by x 1] =( 1 ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 11-x-...

Question

${lim}_{x \to 1} \frac{1 - x^{- \frac{1}{3}}}{1 - x^{- \frac{2}{3}}}$

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Solution

${lim}_{x \to 1} \frac{- (x^{- \frac{1}{3}} - 1)}{- (x^{- \frac{2}{3}} - 1)}$

$= {lim}_{x \to 1} \frac{x^{- \frac{1}{3}} - 1^{- \frac{1}{3}}}{x^{- \frac{2}{3}} - 1^{- \frac{2}{3}}}$

$= {lim}_{x \to 1} \frac{x^{- \frac{1}{3}} - 1^{- \frac{1}{3}}}{\frac{x^{- \frac{2}{3}} - 1^{- \frac{2}{3}}}{x - 1}}$

[Dividing numerator and drominator by x-1]

$= \frac{(- \frac{1}{3}) (1)^{- \frac{1}{3} - 1}}{(- \frac{2}{3}) (1)^{- \frac{2}{3} - 1}}$

$[{lim}_{x \to a} \frac{x^{n} - a^{n}}{x - a} = a^{n - 1}, n \to Q]$

$= \frac{1}{3} \times \frac{3}{2} = \frac{1}{2}$

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Similar questions

{lim}_{x \to 1} \frac{1 - x^{- 1 / 3}}{1 - x^{- 2 / 3}}

Q. Evaluate :

lim x \to 1 \frac{1 - x^{- 1 / 3}}{1 - x^{- 2 / 3}}

lim x \to 1 \frac{1 - x^{- 2 / 3}}{1 - x^{- 1 / 3}}

is equal to

\lim_{x \to 1} \frac{1 - x^{- 1 / 3}}{1 - x^{- 2 / 3}}

lim x \to 1 (\frac{1}{1 - x} - \frac{3}{1 - x^{3}})

is equal to

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limx→11−x−131−x−23

limx→1−(x−13−1)−(x−23−1) =limx→1x−13−1−13x−23−1−23 =limx→1x−13−1−13x−23−1−23x−1 [Dividing numerator and drominator by x-1] =(−13)(1)−13−1(−23)(1)−23−1 [limx→axn−anx−a=an−1,n→Q] =13×32=12

${lim}_{x \to 1} \frac{1 - x^{- \frac{1}{3}}}{1 - x^{- \frac{2}{3}}}$