a piecewise function is given. use properties of limits to find the indicated limits, or state that a limit…

a piecewise function is given. use properties of limits to find the indicated limits, or state that a limit does not exist. f(x) = { (x² - 49)/(x - 7) x≠7; 1 x = 7. a. select the correct choice below and fill in any answer boxes in your choice. a. lim f(x) = 14 (type an integer or a simplified fraction.) x→7⁻ b. the limit does not exist. b. select the correct choice below and fill in any answer boxes in your choice. a. lim f(x) = 14 (type an integer or a simplified fraction.) x→7⁺ b. the limit does not exist. c. select the correct choice below and fill in any answer boxes in your choice. a. lim f(x) = □ (type an integer or a simplified fraction.) x→7 b. the limit does not exist.

a piecewise function is given. use properties of limits to find the indicated limits, or state that a limit does not exist. f(x) = { (x² - 49)/(x - 7) x≠7; 1 x = 7. a. select the correct choice below and fill in any answer boxes in your choice. a. lim f(x) = 14 (type an integer or a simplified fraction.) x→7⁻ b. the limit does not exist. b. select the correct choice below and fill in any answer boxes in your choice. a. lim f(x) = 14 (type an integer or a simplified fraction.) x→7⁺ b. the limit does not exist. c. select the correct choice below and fill in any answer boxes in your choice. a. lim f(x) = □ (type an integer or a simplified fraction.) x→7 b. the limit does not exist.

Answer

Explanation:

Step1: Simplify the function for $x\neq7$

For $x\neq7$, $f(x)=\frac{x^{2}-49}{x - 7}=\frac{(x + 7)(x - 7)}{x - 7}=x + 7$.

Step2: Find the left - hand limit as $x\to7^{-}$

We use the simplified form $f(x)=x + 7$. Substitute $x$ values approaching 7 from the left into $x + 7$. $\lim_{x\to7^{-}}f(x)=\lim_{x\to7^{-}}(x + 7)=7+7 = 14$.

Step3: Find the right - hand limit as $x\to7^{+}$

Using the simplified form $f(x)=x + 7$, substitute $x$ values approaching 7 from the right into $x + 7$. $\lim_{x\to7^{+}}f(x)=\lim_{x\to7^{+}}(x + 7)=7+7 = 14$.

Step4: Find the two - sided limit as $x\to7$

Since $\lim_{x\to7^{-}}f(x)=14$ and $\lim_{x\to7^{+}}f(x)=14$, then $\lim_{x\to7}f(x)=14$.

Answer:

a. A. $\lim_{x\to7^{-}}f(x)=14$ b. A. $\lim_{x\to7^{+}}f(x)=14$ c. A. $\lim_{x\to7}f(x)=14$