Question

let ( f(x) = \begin{cases} x^2 + 4, & x < -4 \\sqrt{x + 4}, & x geq -4 end{cases} ). compute the following limits or state that they do not exist.

a. ( lim_{x \to -4^-} f(x) )

b. ( lim_{x \to -4^+} f(x) )

c. ( lim_{x \to -4} f(x) )

a. compute the limit of ( lim_{x \to -4^-} f(x) ) or state that it does not exist. select the correct choice below and, if necessary, fill in the answer box to complete your choice

• a. ( lim_{x \to -4^-} f(x) = 20 ) (simplify your answer.)
• b. the limit does not exist.

b. compute the limit of ( lim_{x \to -4^+} f(x) ) or state that it does not exist. select the correct choice below and, if necessary, fill in the answer box to complete your choice

• a. ( lim_{x \to -4^+} f(x) = square ) (simplify your answer.)
• b. the limit does not exist.

Explanation:

Part b:

Step1: Identify the function for \( x \to -4^+ \)

For \( x \geq -4 \), the function is \( f(x) = \sqrt{x + 4} \).

Step2: Substitute \( x = -4 \) into the function

Compute \( \lim_{x \to -4^+} \sqrt{x + 4} \) by substituting \( x = -4 \): \( \sqrt{-4 + 4} = \sqrt{0} = 0 \).

Step1: Recall left and right limits

We found \( \lim_{x \to -4^-} f(x) = 20 \) (from part a) and \( \lim_{x \to -4^+} f(x) = 0 \) (from part b).

Step2: Compare left and right limits

Since \( 20
eq 0 \), the two - sided limit \( \lim_{x \to -4} f(x) \) does not exist.

Answer:

A. \( \lim_{x \to -4^+} f(x) = 0 \)

Part c (if needed, assuming we check the overall limit):