find the limit as x approaches 1 for the following function (y value)\n17. $f(x)=\frac{(x - 1)(x + 5)}{(x…

Explanation:

Step1: Simplify the function

Since $x\neq1$ when finding the limit, we can cancel out the common factor $(x - 1)$ in the numerator and denominator. So $f(x)=\frac{(x - 1)(x + 5)}{(x - 1)(x + 2)}=\frac{x + 5}{x + 2}$ for $x\neq1$.

Step2: Find the left - hand limit

The left - hand limit as $x\to1^{-}$ of $y = f(x)$ is $\lim_{x\to1^{-}}\frac{x + 5}{x + 2}$. Substitute $x = 1$ into $\frac{x + 5}{x + 2}$, we get $\frac{1+5}{1 + 2}=\frac{6}{3}=2$.

Step3: Find the right - hand limit

The right - hand limit as $x\to1^{+}$ of $y = f(x)$ is $\lim_{x\to1^{+}}\frac{x + 5}{x + 2}$. Substitute $x = 1$ into $\frac{x + 5}{x + 2}$, we get $\frac{1+5}{1 + 2}=\frac{6}{3}=2$.

Answer:

Left: 2 Right: 2