https://file.aiquickdraw.com/mathbot/file/1727397704483ue542zse.jpg

We are asked to find \(\lim_{{x \to 4}} f(x)\), given the following piecewise function:

\[
f(x) =
\begin{cases}
2x, & x > 4 \\
5, & x = 4 \\
x^2 - 8, & x < 4
\end{cases}
\]

### Step 1: Evaluate the Left-Hand Limit \(\lim_{{x \to 4^-}} f(x)\)
This is the limit of \(f(x)\) as \(x\) approaches 4 from the left (\(x < 4\)).

For \(x < 4\), the function is \(f(x) = x^2 - 8\). So, we calculate:
\[
\lim_{{x \to 4^-}} f(x) = \lim_{{x \to 4^-}} (x^2 - 8) = 4^2 - 8 = 16 - 8 = 8
\]

### Step 2: Evaluate the Right-Hand Limit \(\lim_{{x \to 4^+}} f(x)\)
This is the limit of \(f(x)\) as \(x\) approaches 4 from the right (\(x > 4\)).

For \(x > 4\), the function is \(f(x) = 2x\). So, we calculate:
\[
\lim_{{x \to 4^+}} f(x) = \lim_{{x \to 4^+}} 2x = 2(4) = 8
\]

### Step 3: Check if the Two-Sided Limit Exists
Since both the left-hand limit and the right-hand limit are equal to 8, the two-sided limit exists. Therefore:
\[
\lim_{{x \to 4}} f(x) = 8
\]

Thus, the limit exists and its value is 8.

Would you like more details or have any questions? 

Here are five follow-up questions that you might find useful:
1. What if the left-hand and right-hand limits were not equal? What would the conclusion be?
2. How does the value of \(f(4)\) affect the limit \(\lim_{{x \to 4}} f(x)\)?
3. What does it mean for a limit to "not exist" in a formal sense?
4. Can a piecewise function have a continuous limit at a boundary point like \(x = 4\)?
5. How do you compute limits for more complicated piecewise functions?

**Tip:** When calculating limits for piecewise functions, always compute the left-hand and right-hand limits separately before drawing any conclusions.

Given f(x) = {2x, x > 4; 5, x = 4; x^2 - 8, x < 4}, find lim(x→4) f(x) if it exists.

Learn how to find the limit of a piecewise function at x = 4, covering left-hand and right-hand limits, and applying the two-sided limit theorem. This example is suitable for high school students in Grades 11-12 studying calculus or pre-calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation