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 bb such that the function f(x)f(x) is continuous at x=0x = 0. 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{4x

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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