Math Problem Statement

Suppose lim(x → 2) ((3f(x) - 1) / (x - 3)) equals 4. What is lim(x → 2) f(x)?

Solution

Given that:

limx2(3f(x)1x3)=4,\lim_{x \to 2} \left( \frac{3f(x) - 1}{x - 3} \right) = 4,

we need to determine:

limx2f(x).\lim_{x \to 2} f(x).

Let's analyze this problem step-by-step.

Step 1: Understanding the Limit Condition

The limit of a function as x2x \to 2 is given by:

limx2(3f(x)1x3)=4.\lim_{x \to 2} \left( \frac{3f(x) - 1}{x - 3} \right) = 4.

This implies that as xx approaches 2, the value of 3f(x)1x3\frac{3f(x) - 1}{x - 3} approaches 4. To avoid an indeterminate form, the numerator 3f(x)13f(x) - 1 must balance with the denominator x3x - 3. As x2x \to 2, the denominator x3x - 3 approaches 1-1.

Step 2: Setting Up the Equation

If xx approaches 2, then:

limx23f(2)123=4.\lim_{x \to 2} \frac{3f(2) - 1}{2 - 3} = 4.

Simplifying the denominator, we get:

3f(2)11=4.\frac{3f(2) - 1}{-1} = 4.

Step 3: Solving for f(2)f(2)

Simplifying:

3f(2)1=4.3f(2) - 1 = -4.

Adding 1 to both sides:

3f(2)=3.3f(2) = -3.

Dividing by 3:

f(2)=1.f(2) = -1.

Final Answer

limx2f(x)=1.\lim_{x \to 2} f(x) = -1.

Would you like more details or have any further questions?

Related Questions

  1. How can you verify if a limit exists for a given function?
  2. What happens if f(2)f(2) was undefined? How would the result change?
  3. How do you apply L'Hospital's Rule to solve limits like this?
  4. How would the result change if the limit was limx3\lim_{x \to 3}?
  5. Can we generalize this method for other types of rational limits?

Tip

When dealing with limits of rational functions, always check if the numerator and denominator both approach zero or infinity to decide if L'Hospital's Rule can be applied.

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Math Problem Analysis

Mathematical Concepts

Limits
Functions
Algebra

Formulas

Limit formula: lim(x → c) f(x)
Solving for f(x) using limits

Theorems

Limit of a function

Suitable Grade Level

Grades 11-12