Question

let \\( f(x) = \

$$\begin{cases} x^2 + 19, & x < -19 \\\\ \\sqrt{x + 19}, & x \\ge -19 \\end{cases}$$

\\). compute the following limits or state that they do not exist.

a. \\( \lim_{x \to -19^-} f(x) \\)
b. \\( \lim_{x \to -19^+} f(x) \\)
c. \\( \lim_{x \to -19} f(x) \\)

a. \\( \lim_{x \to -19^-} f(x) = 380 \\) (simplify your answer.)

b. compute the limit of \\( \lim_{x \to -19^+} 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 -19^+} f(x) = \\) (simplify your answer.)
b. the limit does not exist.

Explanation:

Identify the relevant piece of the function

$$ \text{For } x \to -19^+, \quad x > -19 \implies f(x) = \sqrt{x + 19} $$

Evaluate the limit using direct substitution

$$ \lim_{x \to -19^+} f(x) = \lim_{x \to -19^+} \sqrt{x + 19} = \sqrt{-19 + 19} = \sqrt{0} = 0 $$

Select the correct multiple-choice option

$$ \lim_{x \to -19^+} f(x) = 0 \implies \text{Option A is correct with a value of 0.} $$

Answer:

• (A) \(\lim_{x \to -19^+} f(x) = 0\) (Correct answer)
• (B) The limit does not exist.