https://file.aiquickdraw.com/mathbot/file/1726281315319rul3tb4g.jpg

The problem in the image asks to determine the limit of a piecewise function \( f(x) \) as \( x \to 4^- \). The function is given as:

\[
f(x) = \begin{cases} 
-3x - 7 & \text{for } x < 4 \\
4x - 6 & \text{for } x \geq 4
\end{cases}
\]

We are asked to find the left-hand limit, that is, \( \lim_{x \to 4^-} f(x) \). To do this, we will use the expression for \( f(x) \) when \( x < 4 \), which is \( f(x) = -3x - 7 \).

### Step-by-step solution:

1. **Substitute \( x = 4 \) into the equation for \( x < 4 \):**

\[
f(x) = -3x - 7
\]

Substituting \( x = 4 \):

\[
f(4^-) = -3(4) - 7 = -12 - 7 = -19
\]

Therefore, the left-hand limit is \( \lim_{x \to 4^-} f(x) = -19 \).

### Conclusion:
The correct answer is **-19**.

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Would you like more details or have any other questions?

Here are 5 related questions you might find useful:
1. How do we find the right-hand limit \( \lim_{x \to 4^+} f(x) \)?
2. What happens if the left-hand and right-hand limits are not equal?
3. How do we determine whether the function is continuous at \( x = 4 \)?
4. Can you explain what a piecewise function is in more depth?
5. What are the conditions for the existence of a limit at a point?

**Tip:** When working with piecewise functions, always ensure that you use the correct expression for the function based on the direction of the limit (left or right).

Determine the limit by sketching an appropriate graph. f(x) = -3x - 7 for x < 4 and 4x - 6 for x >= 4. What is the limit as x approaches 4 from the left?

Learn how to determine the left-hand limit of the piecewise function f(x) = -3x - 7 (for x < 4) and f(x) = 4x - 6 (for x >= 4) as x approaches 4. The solution explains key concepts like limits and piecewise functions, suitable for students in Grades 10-12.

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