Math Problem Statement
give me the righg answer
If f(x) is continuous at x = 0, find the value a such that f(x) is continuous at x = 0, where f(x) = (√(2x + 1) - 1) / sin(3x) for x ≠ 0, and f(x) = a for x = 0.
Solution
To solve this problem, we need to determine the value of such that the function is continuous at .
The given function is:
\begin{cases} \frac{\sqrt{2x + 1} - 1}{\sin(3x)} & \text{for} \ x \neq 0, \\ a & \text{for} \ x = 0. \end{cases}$$ ### Continuity Condition: For $$f(x)$$ to be continuous at $$x = 0$$, the following condition must hold: $$\lim_{x \to 0} f(x) = f(0) = a.$$ We will first find the limit of $$f(x)$$ as $$x \to 0$$ for $$x \neq 0$$, and then set that equal to $$a$$. ### Step 1: Find the limit as $$x \to 0$$ for $$x \neq 0$$ We need to evaluate: $$\lim_{x \to 0} \frac{\sqrt{2x + 1} - 1}{\sin(3x)}.$$ Using the Taylor expansion around $$x = 0$$ for small $$x$$: - $$\sqrt{2x + 1} \approx 1 + x$$, and - $$\sin(3x) \approx 3x$$. Substitute these approximations: $$\lim_{x \to 0} \frac{\sqrt{2x + 1} - 1}{\sin(3x)} \approx \lim_{x \to 0} \frac{x}{3x} = \frac{1}{3}.$$ ### Step 2: Set the limit equal to $$a$$ For continuity, we set this limit equal to $$a$$: $$a = \frac{1}{3}.$$ Thus, the correct value of $$a$$ is $$\frac{1}{3}$$, which corresponds to option **c)**. --- Do you want more details or have any questions? ### Related questions: 1. How can you prove a function is continuous using limits? 2. What is the Taylor series for $$\sin(x)$$ near zero? 3. Can you explain how to handle indeterminate forms when computing limits? 4. What does it mean for a function to be continuous at a point? 5. How does the small-angle approximation help simplify limit problems? ### Tip: When faced with a limit involving square roots and trigonometric functions, try using series expansions or L'Hôpital's Rule to simplify the expression.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Taylor Series
Trigonometry
Formulas
Taylor Expansion of √(2x + 1)
sin(3x) ≈ 3x as x → 0
Theorems
Continuity at a point
Limit definition of continuity
Suitable Grade Level
Grade 12 or Introductory Calculus
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