Math Problem Statement

Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x) = (x^2 - x) / (x^2 - 1) if x ≠ 1, and f(x) = 1 if x = 1. Determine if the function is discontinuous at a = 1.

Solution

The problem asks to explain why the function is discontinuous at the given number a=1a = 1 and sketch the graph of the function. The function is given as:

\frac{x^2 - x}{x^2 - 1} & \text{if } x \neq 1 \\ 1 & \text{if } x = 1 \end{cases}$$ ### Step-by-Step Solution #### 1. Discontinuity at $$x = 1$$ To determine whether the function is continuous at $$x = 1$$, we need to check the following three conditions: 1. **The function is defined at $$x = 1$$:** $$f(1) = 1$$ So, the function is defined at $$x = 1$$. 2. **The limit as $$x$$ approaches 1 exists:** We need to evaluate the limit as $$x \to 1$$ for the expression $$\frac{x^2 - x}{x^2 - 1}$$ when $$x \neq 1$$. We simplify the expression: $$\frac{x^2 - x}{x^2 - 1} = \frac{x(x - 1)}{(x - 1)(x + 1)} = \frac{x}{x + 1}, \quad \text{for } x \neq 1$$ Now, we find the limit as $$x$$ approaches 1: $$\lim_{x \to 1} \frac{x}{x + 1} = \frac{1}{1 + 1} = \frac{1}{2}$$ So, the limit as $$x \to 1$$ is $$\frac{1}{2}$$. 3. **The limit must equal the function's value at $$x = 1$$:** We have: $$\lim_{x \to 1} f(x) = \frac{1}{2} \quad \text{and} \quad f(1) = 1$$ Since $$\frac{1}{2} \neq 1$$, the function is **discontinuous** at $$x = 1$$. #### Conclusion The function is discontinuous at $$x = 1$$ because the limit of $$f(x)$$ as $$x \to 1$$ does not equal the value of the function at $$x = 1$$. ### Graph Sketch The function is piecewise-defined. For $$x \neq 1$$, it behaves like $$\frac{x}{x+1}$$, which is a rational function. There will be a removable discontinuity at $$x = 1$$, where the value jumps from $$\frac{1}{2}$$ (the limit) to $$1$$ (the actual function value). Would you like further details or a more in-depth explanation on any step? ### Relative Questions 1. What are the conditions for a function to be continuous at a point? 2. How can you determine the type of discontinuity for a given function? 3. How does simplifying rational expressions help in evaluating limits? 4. What is the graphical significance of removable discontinuities? 5. How can piecewise functions be represented graphically? ### Tip: Always check for removable discontinuities by factoring and simplifying rational expressions before evaluating limits.

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Discontinuity
Rational Functions
Piecewise Functions

Formulas

f(x) = (x^2 - x) / (x^2 - 1) for x ≠ 1
f(x) = 1 for x = 1
lim_{x → 1} (x^2 - x) / (x^2 - 1) = 1/2

Theorems

Definition of Continuity
Limit of Rational Functions

Suitable Grade Level

Grade 11-12 or College Calculus I