a) lim_(x rarr2)(x^(2)+x-6)/(x-2)

Factorize the numerator: Simplify the expression in the numerator: (x^2 + x - 6) can be factored into (x - 2)(x + 3). Calculation: x^2 + x - 6 = x^2 - 2x + 3x - 6 = (x(x - 2) + 3(x - 2)) = (x - 2)(x + 3).Substitute factored form: Substitute the factored form back into the original limit expression: lim_(x rarr2)((x - 2)(x + 3))/(x - 2). Calculation: (x - 2) terms cancel out, leaving lim_(x rarr2)(x + 3).Evaluate the limit: Evaluate the limit of the simplified expression: lim_(x rarr2)(x + 3) = 2 + 3. Calculation: 2 + 3 = 5.

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\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}

Factorize the numerator: Simplify the expression in the numerator: $\newline$ $(x^2 + x - 6)$ can be factored into $(x - 2)(x + 3)$ . $\newline$ Calculation: $x^2 + x - 6 = x^2 - 2x + 3x - 6 = (x(x - 2) + 3(x - 2)) = (x - 2)(x + 3)$ .
Substitute factored form: Substitute the factored form back into the original limit expression: $\lim_{x \rightarrow 2}\frac{(x - 2)(x + 3)}{x - 2}$ . Calculation: $(x - 2)$ terms cancel out, leaving $\lim_{x \rightarrow 2}(x + 3)$ .
Evaluate the limit: Evaluate the limit of the simplified expression: $\lim_{x \rightarrow 2}(x + 3) = 2 + 3$ . Calculation: $2 + 3 = 5$ .