Math Problem Statement
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?
Solution
The problem in the image asks to determine the limit of a piecewise function as . The function is given as:
-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$$. ### Conclusion: The correct answer is **-19**. --- Would you like more details or have any other questions? Here are 5 related questions you might find useful: 1. How do we find the right-hand limit $$\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).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
lim(x→4^-) f(x) = -3x - 7
Theorems
Limit Theorem
Suitable Grade Level
Grades 10-12