https://file.aiquickdraw.com/mathbot/file/1726393054317ujk70yv6.jpg

The function \( f(x) \) is defined as follows:

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

### 1. \( \lim_{x \to -1^+} f(x) \)

For this limit, we consider the function's behavior as \( x \to -1 \) from the right (i.e., for values of \( x \geq -1 \)). The function for \( x \geq -1 \) is \( f(x) = \sqrt{x + 1} \).

\[
\lim_{x \to -1^+} f(x) = \lim_{x \to -1^+} \sqrt{x + 1} = \sqrt{-1 + 1} = \sqrt{0} = 0
\]

Thus, \( \lim_{x \to -1^+} f(x) = 0 \).

### 2. \( \lim_{x \to -1^-} f(x) \)

For this limit, we consider the function's behavior as \( x \to -1 \) from the left (i.e., for values of \( x < -1 \)). The function for \( x < -1 \) is \( f(x) = x^2 + 1 \).

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

Thus, \( \lim_{x \to -1^-} f(x) = 2 \).

### Summary of Limits

1. \( \lim_{x \to -1^+} f(x) = 0 \)
2. \( \lim_{x \to -1^-} f(x) = 2 \)

Since the left-hand and right-hand limits do not agree, the overall limit \( \lim_{x \to -1} f(x) \) does not exist.

Do you want further details on any step? Here are 5 related questions:

1. What is the definition of one-sided limits?
2. How does the behavior of piecewise functions affect limits?
3. Can you explain why the limit does not exist if the one-sided limits are unequal?
4. How would the limit change if the function were continuous at \( x = -1 \)?
5. Can we use L'Hopital's Rule to solve limits involving square roots?

**Tip:** When dealing with piecewise functions, always check the left-hand and right-hand limits separately before concluding about the overall limit.

Let f(x) = {x^2 + 1 if x < -1, sqrt(x + 1) if x >= -1}. Compute the following limits or state that they do not exist: 1) lim x->-1^+ f(x), 2) lim x->-1^- f(x).

Learn how to solve a limit problem involving a piecewise function where one piece is a square root function and the other is a quadratic. Explore how the one-sided limits differ and determine if the overall limit exists. Suitable 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