_ x 1 1-x^ -2 3 1-x^ 1 3 is equal to - Vidyadip

MCQ

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

✓
2
B
1
C
$\frac{2}{3}$
D
$\frac{1}{3}$

✓

Answer

Correct option: A.

2

A)
$\lim _{x \rightarrow 1} \frac{1-x^{-\frac{2}{3}}}{1-x^{-\frac{1}{3}}}=\lim _{x \rightarrow 1} \frac{1^2-\left(x^{-\frac{1}{3}}\right)^2}{1-x^{\frac{1}{3}}}$
$=\lim _{x \rightarrow 1} \frac{\left(1+x^{-\frac{1}{3}}\right)\left(1-x^{-\frac{1}{3}}\right)}{1-x^{-\frac{1}{3}}}$
$=\lim _{x \rightarrow 1}\left(1+x^{-1 / 3}\right)=2$
Alternate method:
Apply L- Hospital's Rule.

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