Math Problem Statement
mple: Show that f(x) is discontinuous at a = 2 where, f(x)= x^ 2 -x-2 x-2 & if x ne2\ 1&x=2
Solution
We are given the piecewise function:
\frac{x^2 - x - 2}{x - 2} & \text{if } x \neq 2 \\ 1 & \text{if } x = 2 \end{cases}$$ We are tasked with showing that $$f(x)$$ is discontinuous at $$a = 2$$. ### Step 1: Simplifying the function for $$x \neq 2$$ The function for $$x \neq 2$$ is given as: $$f(x) = \frac{x^2 - x - 2}{x - 2}$$ First, factor the quadratic expression in the numerator: $$x^2 - x - 2 = (x - 2)(x + 1)$$ Thus, the function becomes: $$f(x) = \frac{(x - 2)(x + 1)}{x - 2}, \quad \text{for } x \neq 2$$ Since $$x \neq 2$$, we can cancel the common factor $$x - 2$$: $$f(x) = x + 1, \quad \text{for } x \neq 2$$ Thus, for $$x \neq 2$$, we have $$f(x) = x + 1$$. ### Step 2: Checking the limit of $$f(x)$$ as $$x \to 2$$ Next, we calculate the limit of $$f(x)$$ as $$x$$ approaches 2: $$\lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 1) = 2 + 1 = 3$$ ### Step 3: Comparing the limit and the function value at $$x = 2$$ We know that: $$f(2) = 1$$ However, from Step 2, we found that: $$\lim_{x \to 2} f(x) = 3$$ ### Step 4: Conclusion of discontinuity Since the limit of $$f(x)$$ as $$x \to 2$$ is 3, but $$f(2) = 1$$, the function is not continuous at $$x = 2$$. Therefore, $$f(x)$$ is **discontinuous at $$x = 2$$**. --- Do you want more details on this process or have any questions? ### 5 Relative Questions: 1. What are the conditions for continuity of a function at a point? 2. How can we determine if a function is removable discontinuous at a point? 3. Can you explain the difference between a removable discontinuity and a jump discontinuity? 4. How would the function change if $$f(2)$$ were redefined to match the limit? 5. Can you provide an example of a function that is continuous everywhere? **Tip**: Always check both the limit and the actual function value to confirm if a function is continuous at a point.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Piecewise Functions
Algebraic Factorization
Formulas
f(x) = (x^2 - x - 2) / (x - 2) if x ≠ 2
Limit as x → a
Quadratic Factorization: x^2 - x - 2 = (x - 2)(x + 1)
Theorems
Limit Theorem
Definition of Continuity
Discontinuity at a Point
Suitable Grade Level
Grades 10-12
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