Question

simplify the function algebraically and find the limit.
\\(\lim_{x\to2}\frac{x^{2}+5x - 14}{x^{2}-4x + 4}\\)
simplify the given limit.
\\(\lim_{x\to2}\frac{x^{2}+5x - 14}{x^{2}-4x + 4}=\lim_{x\to2}(\text{type an expression using }x\text{ as the variable. simplify your answer.})\\)

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}+5x - 14$ can be factored as $(x + 7)(x-2)$ using the formula $x^{2}+(a + b)x+ab=(x + a)(x + b)$ where $a = 7$ and $b=-2$. The denominator $x^{2}-4x + 4$ can be factored as $(x - 2)^{2}$ using the perfect - square formula $(a - b)^{2}=a^{2}-2ab + b^{2}$ with $a=x$ and $b = 2$. So, $\lim_{x
ightarrow2}\frac{x^{2}+5x - 14}{x^{2}-4x + 4}=\lim_{x
ightarrow2}\frac{(x + 7)(x - 2)}{(x - 2)^{2}}$.

Step2: Simplify the function

Cancel out the common factor $(x - 2)$ (since $x
eq2$ when taking the limit) in the numerator and denominator. We get $\lim_{x
ightarrow2}\frac{x + 7}{x - 2}$.

The limit $\lim_{x
ightarrow2}\frac{x + 7}{x - 2}$ does not exist because as $x
ightarrow2^{+}$, $\frac{x + 7}{x - 2}
ightarrow+\infty$ and as $x
ightarrow2^{-}$, $\frac{x + 7}{x - 2}
ightarrow-\infty$.

Answer:

The limit does not exist.