To determine the limit of a piecewise function, we need to analyze the function's behavior as it approaches a particular point, considering each piece of the function that might be relevant near that point. Here’s a step-by-step guide to finding limits for piecewise functions:

### Step-by-Step Approach for Finding Limits of Piecewise Functions

1. **Identify the point at which the limit is being evaluated**:
   - Let’s denote this point as \( x = a \). The value of \( a \) is where you want to find the limit of the piecewise function.

2. **Examine the definition of the piecewise function**:
   - Write down the different pieces of the function and their corresponding intervals. For example, a piecewise function might look like this:
     \[
     f(x) = 
     \begin{cases} 
      f_1(x) & \text{if } x < a \\
      f_2(x) & \text{if } x = a \\
      f_3(x) & \text{if } x > a 
     \end{cases}
     \]

3. **Determine the left-hand limit** (\(\lim_{x \to a^-} f(x)\)):
   - Use the piece of the function defined for values **less than** \( a \) (typically \( f_1(x) \)). Substitute values approaching \( a \) from the left into this piece and find the limit:
     \[
     \lim_{x \to a^-} f(x) = \lim_{x \to a^-} f_1(x)
     \]
   - This involves plugging in values just slightly less than \( a \) into \( f_1(x) \).

4. **Determine the right-hand limit** (\(\lim_{x \to a^+} f(x)\)):
   - Use the piece of the function defined for values **greater than** \( a \) (typically \( f_3(x) \)). Substitute values approaching \( a \) from the right into this piece and find the limit:
     \[
     \lim_{x \to a^+} f(x) = \lim_{x \to a^+} f_3(x)
     \]
   - This involves plugging in values just slightly greater than \( a \) into \( f_3(x) \).

5. **Compare the left-hand and right-hand limits**:
   - If \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\) are **equal**, then the two-sided limit exists and is equal to that common value:
     \[
     \lim_{x \to a} f(x) = \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)
     \]
   - If the left-hand limit and right-hand limit are **not equal**, then the limit does not exist at \( x = a \).

### Example:

Suppose we have the piecewise function:
\[
f(x) =
\begin{cases}
2x + 1 & \text{if } x < 3 \\
5 & \text{if } x = 3 \\
x^2 - 4 & \text{if } x > 3
\end{cases}
\]

Find \(\lim_{x \to 3} f(x)\).

**Step 1**: \( a = 3 \).

**Step 2**: Identify the relevant pieces of the function:
   - For \( x < 3 \), use \( f(x) = 2x + 1 \).
   - For \( x = 3 \), use \( f(x) = 5 \).
   - For \( x > 3 \), use \( f(x) = x^2 - 4 \).

**Step 3**: Find the left-hand limit:
   \[
   \lim_{x \to 3^-} (2x + 1) = 2(3) + 1 = 7
   \]

**Step 4**: Find the right-hand limit:
   \[
   \lim_{x \to 3^+} (x^2 - 4) = 3^2 - 4 = 5
   \]

**Step 5**: Compare the limits:
   - \(\lim_{x \to 3^-} f(x) = 7\)
   - \(\lim_{x \to 3^+} f(x) = 5\)

Since these two limits are not equal, \(\lim_{x \to 3} f(x)\) **does not exist**.

Would you like more details or have any questions about this process?

### 5 Related Questions:
1. How do you find limits at points of discontinuity in a piecewise function?
2. What are one-sided limits, and why are they important in evaluating piecewise functions?
3. How can you determine if a piecewise function is continuous at a point?
4. What is the difference between jump discontinuity and removable discontinuity?
5. How can graphing a piecewise function help in finding its limit?

### Tip:
When finding limits of piecewise functions, always pay close attention to which piece is defined on the interval approaching the point, as it dictates which formula you use for left- and right-hand limits.

Discover how to calculate limits for piecewise functions with a detailed step-by-step approach. This guide covers important mathematical concepts like left-hand and right-hand limits and provides an example to illustrate the process. Perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation