find the limits of f(x) as x approaches 2 from the right in this piecewise function, |x+3|, x<2 and x^2+1, x>2

The piecewise function is defined as:

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

We are interested in finding the limit of \( f(x) \) as \( x \) approaches 2 from the right, denoted as \( \lim_{x \to 2^+} f(x) \).

### Step-by-Step Solution:

For \( x > 2 \), the function is defined as \( f(x) = x^2 + 1 \). To find the right-hand limit as \( x \to 2 \):

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

Substitute \( x = 2 \):

\[
\lim_{x \to 2^+} (x^2 + 1) = 2^2 + 1 = 4 + 1 = 5
\]

Thus, the right-hand limit of \( f(x) \) as \( x \to 2^+ \) is 5.

### Conclusion:
\[
\lim_{x \to 2^+} f(x) = 5
\]

Do you want more details or have any questions?

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Here are 5 related questions you can explore:
1. What is the left-hand limit \( \lim_{x \to 2^-} f(x) \)?
2. Is \( f(x) \) continuous at \( x = 2 \)?
3. How do you evaluate limits for functions with absolute values?
4. What happens if \( x = 2 \) for the given piecewise function?
5. Can you graph this piecewise function and analyze its behavior around \( x = 2 \)?

**Tip:** Always check both the right-hand and left-hand limits to determine if a function is continuous at a point.

Explore how to find the right-hand limit of a piecewise function as x approaches 2. This solution involves understanding limits, piecewise functions, and evaluating limits from the right. Suitable for advanced high school students in Grades 11-12.

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