Math Problem Statement
Let π(π₯) be the function such that π(π₯) = { π₯ 2βπ 2 π₯βπ , ππ π₯ β π 0, ππ π₯ = π a. Does π(π) exist? b. Does limπ₯βπ π(π₯) exist? c. Is π(π₯) continuous at π₯ = π?
Solution
Let's break down the given function and address the questions step by step.
We are given:
\begin{cases} \frac{x^2 - b^2}{x - b} & \text{if } x \neq b \\ 0 & \text{if } x = b \end{cases}$$ ### Part a: Does $$g(b)$$ exist? We know that $$g(b) = 0$$ from the definition of the function. Therefore, the value of $$g(b)$$ **does exist**, and it is $$g(b) = 0$$. ### Part b: Does $$\lim_{x \to b} g(x)$$ exist? We need to check if the limit of $$g(x)$$ as $$x$$ approaches $$b$$ exists. To do so, we analyze the expression $$g(x)$$ for $$x \neq b$$: $$g(x) = \frac{x^2 - b^2}{x - b}$$ Notice that $$x^2 - b^2$$ can be factored as $$(x - b)(x + b)$$, so we have: $$g(x) = \frac{(x - b)(x + b)}{x - b}$$ For $$x \neq b$$, the terms $$(x - b)$$ cancel out, and we are left with: $$g(x) = x + b$$ Thus, for $$x \neq b$$, the function simplifies to $$g(x) = x + b$$. Now, we find the limit of this expression as $$x \to b$$: $$\lim_{x \to b} g(x) = \lim_{x \to b} (x + b) = b + b = 2b$$ So, the limit of $$g(x)$$ as $$x \to b$$ is $$2b$$. ### Part c: Is $$g(x)$$ continuous at $$x = b$$? For a function to be continuous at a point, three conditions must be met: 1. The function value at that point must exist. 2. The limit of the function as $$x$$ approaches that point must exist. 3. The limit of the function as $$x$$ approaches that point must be equal to the function value at that point. We already know: - $$g(b) = 0$$. - $$\lim_{x \to b} g(x) = 2b$$. For $$g(x)$$ to be continuous at $$x = b$$, we need the limit to be equal to the function value. Since $$\lim_{x \to b} g(x) = 2b$$ and $$g(b) = 0$$, these values are not equal unless $$b = 0$$. Therefore, **$$g(x)$$ is not continuous at $$x = b$$** unless $$b = 0$$. ### Summary: - a. $$g(b)$$ exists and equals 0. - b. $$\lim_{x \to b} g(x)$$ exists and equals $$2b$$. - c. $$g(x)$$ is not continuous at $$x = b$$ unless $$b = 0$$. Would you like a more detailed explanation of any part? Or would you like me to explore another topic? Here are some related questions: 1. How would the behavior of $$g(x)$$ change if $$g(x)$$ were defined as $$\frac{x^2 - b^2}{x - b}$$ for all values of $$x$$? 2. How do we determine if a limit exists when a function has a piecewise definition? 3. What is the significance of continuity in real-world applications? 4. How do we compute limits for rational functions like this one? 5. What are some conditions under which a function is continuous at a point? **Tip:** When checking limits, always try to simplify expressions before directly substituting the limit value!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Factoring in Algebra
Formulas
Difference of squares: a^2 - b^2 = (a - b)(a + b)
Limit definition: lim xβc f(x) = L if left-hand and right-hand limits exist and are equal
Theorems
Limit Laws
Continuity Definition
Factoring Theorem
Suitable Grade Level
Grades 11-12 / Early College
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