https://file.aiquickdraw.com/mathbot/file/1726140022058ntlmsy7o.jpg

The problem asks to evaluate the limit of a piecewise function as \(x\) approaches 2, and determine whether it exists by evaluating the one-sided limits.

The function is defined as:

\[
f(x) = \begin{cases}
x - 1 & \text{for } x \leq 2 \\
2x - 3 & \text{for } x > 2
\end{cases}
\]

### Steps to solve:

1. **Left-hand limit**: 
   We evaluate the limit as \( x \to 2^- \) (approaching 2 from the left). For \(x \leq 2\), the function is \( f(x) = x - 1 \).
   \[
   \lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} (x - 1) = 2 - 1 = 1
   \]

2. **Right-hand limit**: 
   We evaluate the limit as \( x \to 2^+ \) (approaching 2 from the right). For \(x > 2\), the function is \( f(x) = 2x - 3 \).
   \[
   \lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (2x - 3) = 2(2) - 3 = 4 - 3 = 1
   \]

3. **Comparing the limits**: 
   Since both the left-hand limit and right-hand limit are equal:
   \[
   \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = 1
   \]
   The two-sided limit exists and is equal to 1:
   \[
   \lim_{x \to 2} f(x) = 1
   \]

### Graph of the Function:
The function has two different expressions depending on whether \(x\) is less than, equal to, or greater than 2.

- For \(x \leq 2\), the function is \(f(x) = x - 1\), which is a line with a slope of 1.
- For \(x > 2\), the function is \(f(x) = 2x - 3\), which is a line with a slope of 2.

The graph should show continuity at \(x = 2\), with both pieces meeting at the point \((2, 1)\).

Would you like a more detailed explanation or assistance with graphing this?

Here are 5 related questions to expand on this topic:

1. What happens if the left-hand and right-hand limits are not equal at a point?
2. How would you define continuity of a function at a point in terms of limits?
3. What is the difference between one-sided and two-sided limits?
4. How do you graph piecewise functions in general?
5. Can a piecewise function have a jump discontinuity, and how can you identify it?

**Tip**: When evaluating limits, always check both one-sided limits if the function is piecewise-defined at the point in question.

Graph the function. Determine the limit (if it exists) by evaluating the corresponding one-sided limits. The function is defined as f(x) = {x - 1, x ≤ 2; 2x - 3, x > 2} and we need to find the limit as x approaches 2.

Learn how to evaluate the limit of a piecewise function as x approaches a point by examining one-sided limits. This problem involves understanding limits, continuity, and the behavior of piecewise-defined functions, suitable for high school 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