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

To find the limit of \( f(x) \) as \( x \) approaches 2 from the left, we need to use the expression defined for \( x < 2 \) in the given piecewise function.

The function is given as:

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

Since we are looking for the limit of \( f(x) \) as \( x \to 2^- \) (from the left), we use the expression for \( x < 2 \), which is \( f(x) = x^2 + 1 \).

Now, let's compute the limit:

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

Substituting \( x = 2 \) into the expression \( x^2 + 1 \):

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

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

Would you like details or have any questions about this? Here are five related questions:

1. How do you compute one-sided limits for other types of functions?
2. What happens if the limit from the left and right are not the same?
3. How do absolute value functions behave near critical points like \( x = 2 \)?
4. What would be the limit from the right for this particular function?
5. Can a piecewise function be continuous at the boundary between the pieces?

**Tip:** When dealing with piecewise functions, always check which part of the function applies to the specific direction of the limit (from the left or right).

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

Find the limit of the piecewise function f(x) as x approaches 2 from the left. The problem involves understanding one-sided limits and how to handle different expressions in piecewise functions. Suitable for high school students (Grades 10-12).

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