Questions: f(x) is discontinuous at x=4. (Use a comma to separate answers as needed.) (a) Find lim f(x) for the a-value found above. Select the correct choice below and fill in any answer boxes in your choice. x -> a^- A. lim x -> a^- f(x)= (Type an integer or a simplified fraction.) B. The limit does not exist.

$f(x) is discontinuous at x=4. (Use a comma to separate answers as needed.) (a) Find lim f(x) for the a-value found above. Select the correct choice below and fill in any answer boxes in your choice. x -> a^- A. lim x -> a^- f(x)= (Type an integer or a simplified fraction.) B. The limit does not exist.$

Transcript text: $f(x)$ is discontinuous at $x=4$. (Use a comma to separate answers as needed.) (a) Find $\lim f(x)$ for the a-value found above. Select the correct choice below and fill in any answer boxes in your choice. \[ \mathrm{x} \rightarrow \mathrm{a}^{-} \] A. $\lim _{x \rightarrow a^{-}} f(x)=\square$ (Type an integer or a simplified fraction.) B. The limit does not exist.

Solution

Solution Steps

To solve this problem, we need to evaluate the left-hand limit of the function $ f(x) $ as $ x $ approaches 4. This involves analyzing the behavior of $ f(x) $ as $ x $ gets closer to 4 from the left side. If the function approaches a specific value, that value is the limit; otherwise, the limit does not exist.

Step 1: Define the Function

We are given a piecewise function $ f(x) $ defined as: \[ f(x) = \begin{cases} x^2 & \text{if } x < 4 \\ 0 & \text{if } x \geq 4 \end{cases} \]

Step 2: Evaluate the Left-Hand Limit

To find $ \lim_{x \to 4^{-}} f(x) $, we need to consider the value of $ f(x) $ as $ x $ approaches 4 from the left. For $ x < 4 $, $ f(x) = x^2 $.

Step 3: Calculate the Limit

We calculate the limit: \[ \lim_{x \to 4^{-}} f(x) = \lim_{x \to 4^{-}} x^2 = 4^2 = 16 \]