Math Problem Statement
For what value(s) of b is the function f(x) continuous at x = 3 and does f(x) have a removable discontinuity at x = 3?
Solution
Let's go through the problem step by step.
The function is defined piecewise as:
\begin{cases} -x + b, & \text{if } x < 3 \\ 1, & \text{if } x = 3 \\ -\frac{1}{x - b} + 0, & \text{if } x > 3 \text{ (and } x \neq b) \end{cases}$$ ### Part (a): Continuity at $$x = 3$$ For $$f(x)$$ to be continuous at $$x = 3$$, the left-hand limit, the right-hand limit, and the value of the function at $$x = 3$$ must all be equal. 1. **Left-hand limit as $$x \to 3^-$$**: $$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (-x + b) = -3 + b$$ 2. **Right-hand limit as $$x \to 3^+$$**: $$\lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} \left( -\frac{1}{x - b} \right)$$ However, since we are evaluating the limit as $$x$$ approaches 3, we must carefully consider the value of $$b$$. If $$b = 3$$, the denominator would go to zero, indicating a possible discontinuity. 3. **Value at $$x = 3$$**: $$f(3) = 1$$ For continuity at $$x = 3$$: $$\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3) = 1$$ Thus: $$-3 + b = 1 \quad \Rightarrow \quad b = 4$$ ### Part (b): Removable Discontinuity at $$x = 3$$ For $$f(x)$$ to have a removable discontinuity at $$x = 3$$, the left-hand and right-hand limits must be equal, but they do not necessarily need to equal the value of the function at $$x = 3$$. From part (a), we already know: $$\lim_{x \to 3^-} f(x) = -3 + b$$ $$\lim_{x \to 3^+} f(x) = -\frac{1}{x - b} \quad (\text{for values of } b \neq 3)$$ In this case, we need the left-hand and right-hand limits to match. So: $$-3 + b = 1$$ Therefore, the value of $$b$$ remains $$b = 4$$ to match both limits, and hence, it removes the discontinuity at $$x = 3$$. Would you like further details on these steps or have any other questions? Here are 5 related questions to think about: 1. What if $$b = 3$$? How does the function behave then? 2. What is the significance of a removable discontinuity? 3. Can we calculate the derivative of this function at $$x = 3$$? 4. How do we classify discontinuities besides removable ones? 5. How would the function change if the condition for $$x = 3$$ was removed? **Tip**: Always check both the left and right-hand limits for continuity and discontinuity problems; sometimes, one side may exhibit a different type of behavior.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Removable Discontinuity
Piecewise Functions
Formulas
Limits
Piecewise continuity conditions
Left-hand and right-hand limits
Theorems
Continuity Theorem
Removable Discontinuity Definition
Suitable Grade Level
Grades 11-12
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