https://file.aiquickdraw.com/mathbot/file/172576745828284g8s4k9.jpg

The function \( f(x) \) is given as:

\[
f(x) =
\begin{cases} 
x^2 + 1 & \text{if } x < 1 \\
(x - 3)^2 & \text{if } x \geq 1 
\end{cases}
\]

### (a) Find the following limits:

1. \(\lim\limits_{x \to 1^-} f(x)\)

This is the limit as \(x\) approaches 1 from the left. For \(x < 1\), the function is given by \(f(x) = x^2 + 1\). So we compute:

\[
\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (x^2 + 1) = 1^2 + 1 = 2
\]

2. \(\lim\limits_{x \to 1^+} f(x)\)

This is the limit as \(x\) approaches 1 from the right. For \(x \geq 1\), the function is given by \(f(x) = (x - 3)^2\). So we compute:

\[
\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x - 3)^2 = (1 - 3)^2 = (-2)^2 = 4
\]

### Conclusion:
Since \(\lim_{x \to 1^-} f(x) = 2\) and \(\lim_{x \to 1^+} f(x) = 4\), the two one-sided limits are not equal. Therefore, the overall limit \(\lim_{x \to 1} f(x)\) does not exist (DNE).

---

Would you like further details or explanations? 

Here are five related questions to expand on this topic:

1. What is the definition of a one-sided limit?
2. How does the behavior of a piecewise function affect the existence of a limit at a boundary point?
3. Can a limit exist if the one-sided limits are unequal? Why or why not?
4. How can you graphically interpret one-sided limits?
5. What is the importance of continuity in relation to limits?

**Tip:** When dealing with piecewise functions, always check both one-sided limits at the point of interest to determine if the overall limit exists.

f(x) = { x^2 + 1 if x < 1, (x - 3)^2 if x >= 1}. Find the following limits: lim(x→1^-)f(x), lim(x→1^+)f(x).

Learn how to solve one-sided limits for a piecewise function at x = 1. Understand the mathematical concepts, formulas, and theorems involved. This problem is suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation