Math Problem Statement
Determine whether the following function is continuous at a = 15: f(x) = (x^2 - 225) / (x - 15) if x ≠ 15; f(x) = 1 if x = 15. Use the continuity checklist to justify your answer.
Solution
Let's analyze the function and check its continuity at .
Function Definition:
\frac{x^2 - 225}{x - 15} & \text{if } x \neq 15 \\ 1 & \text{if } x = 15 \end{cases}$$ We need to determine whether this function is continuous at $$x = 15$$. ### Continuity Conditions: For a function to be continuous at $$x = a$$, the following three conditions must hold: 1. **$$f(a)$$ is defined.** 2. **$$\lim_{x \to a} f(x)$$ exists.** 3. **$$\lim_{x \to a} f(x) = f(a)$$.** Let's check each condition. ### Step 1: Is $$f(15)$$ defined? Yes, $$f(15) = 1$$. So, this condition is satisfied. ### Step 2: Does $$\lim_{x \to 15} f(x)$$ exist? For $$x \neq 15$$, $$f(x) = \frac{x^2 - 225}{x - 15}$$. This can be simplified as follows: $$\frac{x^2 - 225}{x - 15} = \frac{(x - 15)(x + 15)}{x - 15}.$$ For $$x \neq 15$$, we can cancel the $$(x - 15)$$ terms: $$f(x) = x + 15 \quad \text{for} \quad x \neq 15.$$ Thus, $$\lim_{x \to 15} f(x) = 15 + 15 = 30$$. So, the limit exists and equals 30. ### Step 3: Is $$\lim_{x \to 15} f(x) = f(15)$$? No, $$f(15) = 1$$, but $$\lim_{x \to 15} f(x) = 30$$. So, this condition is not satisfied. ### Conclusion: The function is **not continuous** at $$x = 15$$ because while $$f(15)$$ is defined and the limit exists, the limit does not equal the value of the function at $$x = 15$$. The correct answer is **B**: - The function is not continuous at $$a = 15$$ because although $$f(15) = 1$$ is defined and $$\lim_{x \to 15} f(x) = 30$$ exists, $$f(15) \neq \lim_{x \to 15} f(x)$$. --- Do you want more details or have any questions? Here are 5 related questions: 1. What happens if the limit of a function at a point equals the function's value at that point? 2. How do you handle piecewise functions when checking continuity? 3. Can a function be discontinuous if the limit exists at a given point? 4. How would the function be modified to make it continuous at $$x = 15$$? 5. What role does factoring play in simplifying limits? **Tip**: When simplifying rational expressions to find limits, always check if cancellation of terms changes the domain of the function!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Factoring
Formulas
lim (x → 15) f(x)
(x^2 - 225) = (x - 15)(x + 15)
Theorems
Continuity Theorem
Limit Theorem
Factoring Techniques in Algebra
Suitable Grade Level
Grade 12 / First-year College (Calculus)