Math Problem Statement
Consider the function f(x) = { (4x^2 - 7x)/3x if x ≠ 0, b if x = 0 }. If f is continuous at x = 0, find the value of b.
Solution
The problem in the image asks to find the value of such that the function is continuous at . The function is given as:
\frac{4x^2 - 7x}{3x}, & \text{if } x \neq 0 \\ b, & \text{if } x = 0 \end{cases}$$ To ensure continuity at $$x = 0$$, the limit of $$f(x)$$ as $$x \to 0$$ must equal the function value at $$x = 0$$, which is $$b$$. Thus, we need to compute: $$\lim_{x \to 0} f(x) = f(0) = b$$ ### Step 1: Simplify $$f(x)$$ for $$x \neq 0$$ The expression for $$f(x)$$ when $$x \neq 0$$ is: $$f(x) = \frac{4x^2 - 7x}{3x}$$ Simplify this: \[ f(x) = \frac{x(4x - 7)}{3x} = \frac{4xAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
Limit of a function as x approaches a point
Simplification of rational functions
Theorems
Limit definition of continuity
Suitable Grade Level
Grades 10-12
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