f(x)= 
⎩
⎨
⎧

  
3x−1
0
2x+5

  
if x<0
if x=0
if x>0

 lim f(x) x approches -3

To find the limit of the function \( f(x) \) as \( x \) approaches \(-3\), we need to determine which piece of the piecewise function applies when \( x \) is close to \(-3\).

Given the piecewise function:

\[
f(x) = 
\begin{cases} 
3x - 1 & \text{if } x < 0 \\
0 & \text{if } x = 0 \\
2x + 5 & \text{if } x > 0 
\end{cases}
\]

Since \(-3\) is less than 0, we use the first piece of the function, \( f(x) = 3x - 1 \).

Now, we calculate the limit:

\[
\lim_{{x \to -3}} f(x) = \lim_{{x \to -3}} (3x - 1)
\]

Substitute \( x = -3 \) into the expression:

\[
\lim_{{x \to -3}} (3(-3) - 1) = 3(-3) - 1 = -9 - 1 = -10
\]

Thus, the limit is:

\[
\lim_{{x \to -3}} f(x) = -10
\]

Would you like more details or have any other questions? Here are some related questions you might find useful:

1. What is the limit as \( x \) approaches 3 for the function \( f(x) \)?
2. How do we determine the limit when \( x \) approaches 0 for the given function?
3. How do we handle limits involving discontinuous functions?
4. Can you explain what happens when the limit approaches a point where the function is undefined?
5. What is the significance of one-sided limits in piecewise functions?

**Tip:** When dealing with piecewise functions, always check which piece of the function applies to the value you're approaching in the limit.

Learn how to calculate the limit of the piecewise function f(x) = { 3x - 1 if x < 0, 0 if x = 0, 2x + 5 if x > 0 } as x approaches -3. This problem involves understanding limits and piecewise functions. Suitable for advanced high school students.

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