Question

sketch the graph of the following function and use it to answer the following questions. let
f(x) = \begin{cases} 9 - 5x, & x < -2 \\ 5x, & -2 leq x < 7 \\ (x - 7)^2, & x geq 7 end{cases}
calculate the following limits. enter undefined if the limit does not exist.

a. (limlimits_{x \to -2^-} f(x) =)

b. (limlimits_{x \to -2^+} f(x) =)

c. (limlimits_{x \to -2} f(x) =)

d. (limlimits_{x \to 7^-} f(x) =)

e. (limlimits_{x \to 7^+} f(x) =)

f. (limlimits_{x \to 7} f(x) =)

Explanation:

Part a: $\boldsymbol{\lim_{x \to -2^-} f(x)}$

Step1: Identify the function for $x < -2$

For $x \to -2^-$, we use the first piece of the piecewise function: $f(x) = 9 - 5x$.

Step2: Substitute $x = -2$ into the function

Substitute $x = -2$ into $9 - 5x$: $9 - 5(-2) = 9 + 10 = 19$.

Step1: Identify the function for $x \geq -2$ (near $-2^+$)

For $x \to -2^+$, we use the second piece of the piecewise function: $f(x) = 5x$.

Step2: Substitute $x = -2$ into the function

Substitute $x = -2$ into $5x$: $5(-2) = -10$.

Step1: Check left and right limits

We found that $\lim_{x \to -2^-} f(x) = 19$ and $\lim_{x \to -2^+} f(x) = -10$.

Step2: Determine if the limit exists

Since the left - hand limit ($19$) and the right - hand limit ($-10$) are not equal, the two - sided limit does not exist.

Answer:

$19$

Part b: $\boldsymbol{\lim_{x \to -2^+} f(x)}$