Find the limit: mathop {lim }limits_{x to 1} | vedclass

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(A) We are given the limit $\mathop {\lim }\limits_{x 	o 1} \left[\frac{x-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}ight]$.First,we factorize the deno

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$2$

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(A) We are given the limit $\mathop {\lim }\limits_{x 	o 1} \left[\frac{x-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}ight]$.First,we factorize the denominators:$x^2 - x = x(x-1)$$x^3 - 3x^2 + 2x = x(x^2 - 3x + 2) = x(x-1)(x-2)$Now,we rewrite the expression:$\frac{x-2}{x(x-1)} - \frac{1}{x(x-1)(x-2)} = \frac{(x-2)^2 - 1}{x(x-1)(x-2)}$$= \frac{x^2 - 4x + 4 - 1}{x(x-1)(x-2)} = \frac{x^2 - 4x + 3}{x(x-1)(x-2)}$$= \frac{(x-3)(x-1)}{x(x-1)(x-2)}$Canceling the common factor $(x-1)$ for $x 
eq 1$:$= \frac{x-3}{x(x-2)}$Now,evaluating the limit as $x 	o 1$:$\mathop {\lim }\limits_{x 	o 1} \frac{x-3}{x(x-2)} = \frac{1-3}{1(1-2)} = \frac{-2}{-1} = 2$.

Find the limit: $\mathop {\lim }\limits_{x \to 1} \left[\frac{x-2}{x^{2}-x}-\frac{1}{x^{3}-3 x^{2}+2 x}\right]$.

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