Math Problem Statement
f(x)= ⎩ ⎨ ⎧
3x−1 0 2x+5
if x<0 if x=0 if x>0
lim f(x) x approches -3
Solution
To find the limit of the function as approaches , we need to determine which piece of the piecewise function applies when is close to .
Given the piecewise function:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Formulas
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Theorems
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Suitable Grade Level
Grades 11-12