Math Problem Statement
Find the constant a such that the function f(x) is continuous on the entire real line.
Solution
To solve this problem, we need to ensure that the given piecewise function is continuous on the entire real line, especially at , where the two pieces of the function meet.
Step 1: Analyze the pieces of the function
The function is defined as:
-\frac{4 \sin x}{x}, & x < 0 \\ a + 7x, & x \geq 0 \end{cases}$$ For the function to be continuous at $$x = 0$$, we require the left-hand limit (as $$x \to 0^-$$) to equal the right-hand limit (as $$x \to 0^+$$) and the value of the function at $$x = 0$$. ### Step 2: Find the left-hand limit as $$x \to 0^-$$ As $$x \to 0^-$$, the left-hand side of the function is: $$\lim_{x \to 0^-} -\frac{4 \sin x}{x}$$ Using the known limit identity: $$\lim_{x \to 0} \frac{\sin x}{x} = 1,$$ we get: $$\lim_{x \to 0^-} -\frac{4 \sin x}{x} = -4.$$ ### Step 3: Find the right-hand limit as $$x \to 0^+$$ For $$x \geq 0$$, the function is $$f(x) = a + 7x$$. The right-hand limit as $$x \to 0^+$$ is: $$\lim_{x \to 0^+} (a + 7x) = a.$$ ### Step 4: Set the limits equal To ensure continuity, the left-hand limit must equal the right-hand limit at $$x = 0$$: $$-4 = a.$$ Thus, the value of $$a$$ is $$-4$$. ### Final Answer: The correct answer is $$\boxed{-4}$$ (Option E). Would you like a detailed explanation of any step or have any further questions? Here are 5 related questions to expand your understanding: 1. How do you define the continuity of a function at a point? 2. What is the significance of using the limit identity $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ in this problem? 3. How do you determine if a piecewise function is continuous at a junction point? 4. How does the right-hand and left-hand limit approach help in solving continuity problems? 5. Can you have a continuous function that is not differentiable at some points? **Tip:** When checking continuity for piecewise functions, always compare the left-hand and right-hand limits at the point where the pieces meet.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
\lim_{x \to 0} \frac{\sin x}{x} = 1
Theorems
Limit Theorem for Sine
Continuity Condition at a Point
Suitable Grade Level
Grades 11-12
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