Find the limit: mathop {lim }limits_{x to 2} | vedclass

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(D) We are given the limit $\mathop {\lim }\limits_{x 	o 2} \frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}$.First,we evaluate the function at $x = 2$:$\frac{2^{2}

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Does not exist

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(D) We are given the limit $\mathop {\lim }\limits_{x 	o 2} \frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}$.First,we evaluate the function at $x = 2$:$\frac{2^{2}-4}{2^{3}-4(2)^{2}+4(2)} = \frac{4-4}{8-16+8} = \frac{0}{0}$.Since it is an indeterminate form,we simplify the expression by factoring:$\frac{x^{2}-4}{x(x^{2}-4x+4)} = \frac{(x-2)(x+2)}{x(x-2)^{2}}$.Canceling the common factor $(x-2)$,we get:$\mathop {\lim }\limits_{x 	o 2} \frac{x+2}{x(x-2)}$.As $x 	o 2$,the numerator approaches $4$ and the denominator approaches $2(0) = 0$.Since the limit results in a non-zero constant divided by zero,the limit does not exist (it approaches infinity).

Find the limit: $\mathop {\lim }\limits_{x \to 2} \left[\frac{x^{2}-4}{x^{3}-4 x^{2}+4 x}\right]$

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