Math Problem Statement
Let g(x)equalsleft brace Start 3 By 2 Matrix 1st Row 1st Column x squared plus 3 x 2nd Column ModifyingAbove if x less than 2 With font size decreased by 5 2nd Row 1st Column a 2nd Column if x equals 2 3rd Row 1st Column 4 x plus 7 2nd Column if x greater than 2 EndMatrix . a. Determine the value of a for which g is continuous from the left at 2. b. Determine the value of a for which g is continuous from the right at 2. c. Is there a value of a for which g is continuous at 2? Question content area bottom Part 1 a. The value of a for which g is continuous from the left at 2 is 0. (Simplify your answer.)
Solution
To answer these questions about the function , we need to analyze the conditions for continuity from the left, right, and at .
The function is defined as follows:
\begin{cases} x^2 + 3x & \text{if } x < 2, \\ a & \text{if } x = 2, \\ 4x + 7 & \text{if } x > 2. \end{cases}$$ ### Part (a): Continuity from the left at $$x = 2$$ For $$g(x)$$ to be continuous from the left at $$x = 2$$, the limit of $$g(x)$$ as $$x$$ approaches 2 from the left must equal $$g(2) = a$$. That is, we need: $$\lim_{x \to 2^-} g(x) = a.$$ For $$x < 2$$, the function is given by $$g(x) = x^2 + 3x$$. We will compute the left-hand limit as $$x \to 2$$: $$\lim_{x \to 2^-} g(x) = \lim_{x \to 2^-} (x^2 + 3x) = (2^2 + 3(2)) = 4 + 6 = 10.$$ For $$g(x)$$ to be continuous from the left at $$x = 2$$, we need $$a = 10$$. Thus, the value of $$a$$ for which $$g(x)$$ is continuous from the left at $$x = 2$$ is $$\boxed{10}$$. ### Part (b): Continuity from the right at $$x = 2$$ For $$g(x)$$ to be continuous from the right at $$x = 2$$, the limit of $$g(x)$$ as $$x$$ approaches 2 from the right must equal $$g(2) = a$$. That is, we need: $$\lim_{x \to 2^+} g(x) = a.$$ For $$x > 2$$, the function is given by $$g(x) = 4x + 7$$. We will compute the right-hand limit as $$x \to 2$$: $$\lim_{x \to 2^+} g(x) = \lim_{x \to 2^+} (4x + 7) = 4(2) + 7 = 8 + 7 = 15.$$ For $$g(x)$$ to be continuous from the right at $$x = 2$$, we need $$a = 15$$. Thus, the value of $$a$$ for which $$g(x)$$ is continuous from the right at $$x = 2$$ is $$\boxed{15}$$. ### Part (c): Continuity at $$x = 2$$ For $$g(x)$$ to be continuous at $$x = 2$$, both the left-hand and right-hand limits must equal $$g(2) = a$$. That is, we need: $$\lim_{x \to 2^-} g(x) = \lim_{x \to 2^+} g(x) = g(2) = a.$$ From Part (a), we found that the left-hand limit is 10, and from Part (b), we found that the right-hand limit is 15. Since these two limits are not equal, there is no single value of $$a$$ that makes $$g(x)$$ continuous at $$x = 2$$. Thus, there is **no value** of $$a$$ for which $$g(x)$$ is continuous at $$x = 2$$. --- Do you want any details or further clarification? Here are 5 related questions: 1. What is the definition of continuity at a point? 2. How can you determine if a piecewise function is continuous at a specific point? 3. What happens if the left-hand and right-hand limits of a function are not equal at a point? 4. How do you find limits for piecewise functions? 5. How would the graph of $$g(x)$$ behave around $$x = 2$$ for different values of $$a$$? **Tip:** When solving problems involving continuity, always check both the left-hand and right-hand limits at the point of interest.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Calculus
Continuity
Limits
Piecewise Functions
Formulas
Limit of a function: lim(x → c) f(x) = L
Continuity at a point: A function f(x) is continuous at x = c if lim(x → c^-) f(x) = f(c) = lim(x → c^+) f(x)
Theorems
Continuity Theorem
Limit Definition
Suitable Grade Level
Grades 11-12
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