Lim x - 11-x2-x-2-x-x3-1

Lim_x → 1 ( 1/x^2+2x x 2 x/x^3 1^3 ) =lim_x → 1 ( 1/x(x+2) (x+2) x/(x 1)(x^2+x+1) ) =lim_x → 1 ( 1/(x+2)(x 1)=x/(x 1)(x^2+x+1) ) =lim_x → 11/(x 1) ( 1/x+2 x/x^2 ...

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Byju's Answer

Standard XII

Mathematics

Factorization Method Form to Remove Indeterminate Form

lim x → 11/x2...

Question

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

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Solution

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

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

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

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

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

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

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

$= {lim}_{x \to 1} \frac{1}{(x - 1)} (\frac{- x + 1}{(x - 1) (x + 2) (x^{2} + x + 1)}) (\frac{0}{0} f o r m)$

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

$= \frac{- 1}{(1 + 2) (1 + 1 + 1)}$

$= \frac{1}{9}$

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limx→1(1x2+x−2−xx3−1)

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