Math Problem Statement
Calculate the limits for the given piecewise function at x = 4 and check if the function is continuous.
Solution
Let's analyze the image and solve the problem. The given function is:
\begin{cases} 4x - 2, & x < 4 \\ 2x + 6, & x \geq 4 \end{cases}$$ We are tasked to evaluate: a) $$\lim_{x \to 4^+} f(x)$$ b) $$\lim_{x \to 4^-} f(x)$$ and check whether $$f(x)$$ is continuous at $$x = 4$$. --- ### Part a) $$\lim_{x \to 4^+} f(x)$$ The right-hand limit uses $$f(x)$$ for $$x \geq 4$$, which is $$f(x) = 2x + 6$$. Substitute $$x = 4$$: $$\lim_{x \to 4^+} f(x) = 2(4) + 6 = 8 + 6 = 14.$$ ### Part b) $$\lim_{x \to 4^-} f(x)$$ The left-hand limit uses $$f(x)$$ for $$x < 4$$, which is $$f(x) = 4x - 2$$. Substitute $$x = 4$$: $$\lim_{x \to 4^-} f(x) = 4(4) - 2 = 16 - 2 = 14.$$ ### Continuity Check To determine if $$f(x)$$ is continuous at $$x = 4$$, we check: 1. $$\lim_{x \to 4^-} f(x) = 14$$ (calculated above). 2. $$\lim_{x \to 4^+} f(x) = 14$$ (calculated above). 3. $$f(4)$$: For $$x = 4$$, $$f(x) = 2(4) + 6 = 14$$. Since $$\lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) = f(4) = 14$$, the function is **continuous** at $$x = 4$$. --- ### Final Answers a) $$\lim_{x \to 4^+} f(x) = 14$$ b) $$\lim_{x \to 4^-} f(x) = 14$$ Continuity: **Yes**, $$f(x)$$ is continuous at $$x = 4$$. --- Would you like a more detailed breakdown or have additional questions? #### Related Questions: 1. What are the conditions for a function to be continuous at a point? 2. How do left-hand and right-hand limits help determine continuity? 3. What happens if $$\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$$? 4. Can a piecewise function always be made continuous? 5. How does one graphically verify continuity for piecewise functions? #### Tip: Always check both the limits and the function value at the point when assessing continuity.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Piecewise Functions
Formulas
Limit formula: \(\lim_{x \to c} f(x)\)
Continuity condition: \(\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = f(c)\)
Theorems
Definition of Continuity
Piecewise Function Analysis
Suitable Grade Level
Grades 11-12