Let's solve problem 1(b) from the document.

### Problem 1(b): Limit

Find the limit:

\[
\lim_{x \to 2} \frac{2x - 4}{\sqrt{3x + 3} - 3}
\]

#### Solution Steps:

1. **Direct Substitution Check**:
   - Substitute \( x = 2 \) into the expression to see if it results in a determinate or indeterminate form.

   \[
   \frac{2(2) - 4}{\sqrt{3(2) + 3} - 3} = \frac{4 - 4}{\sqrt{6 + 3} - 3} = \frac{0}{0}
   \]

   This gives the indeterminate form \( \frac{0}{0} \).

2. **Rationalizing the Denominator**:
   - To eliminate the square root in the denominator, multiply the numerator and denominator by the conjugate of the denominator, \(\sqrt{3x + 3} + 3\):

   \[
   \frac{2x - 4}{\sqrt{3x + 3} - 3} \cdot \frac{\sqrt{3x + 3} + 3}{\sqrt{3x + 3} + 3}
   \]

3. **Simplify the Expression**:
   - The numerator becomes:
   
   \[
   (2x - 4)(\sqrt{3x + 3} + 3)
   \]

   - The denominator becomes:

   \[
   (\sqrt{3x + 3})^2 - 3^2 = (3x + 3) - 9
   \]

4. **Further Simplification**:
   - Numerator: 

   \[
   (2x - 4)(\sqrt{3x + 3} + 3)
   \]

   - Denominator: 

   \[
   3x + 3 - 9 = 3x - 6
   \]

   \[
   = 3(x - 2)
   \]

   - The entire expression now becomes:

   \[
   \frac{(2x - 4)(\sqrt{3x + 3} + 3)}{3(x - 2)}
   \]

5. **Cancel Common Factors**:
   - Notice that \( 2x - 4 = 2(x - 2) \). Thus, we can cancel \( x - 2 \) in the numerator and denominator:

   \[
   \frac{2(x - 2)(\sqrt{3x + 3} + 3)}{3(x - 2)} = \frac{2(\sqrt{3x + 3} + 3)}{3}
   \]

6. **Compute the Limit**:
   - Now, substitute \( x = 2 \):

   \[
   \frac{2(\sqrt{3(2) + 3} + 3)}{3} = \frac{2(\sqrt{6 + 3} + 3)}{3} = \frac{2(\sqrt{9} + 3)}{3}
   \]

   \[
   = \frac{2(3 + 3)}{3} = \frac{2 \cdot 6}{3} = \frac{12}{3} = 4
   \]

**Final Answer**: 

\[
\lim_{x \to 2} \frac{2x - 4}{\sqrt{3x + 3} - 3} = 4
\]

Would you like a detailed explanation of any specific part or have questions about another problem? Here are some other problems to consider:

1. Solve Problem 1(c) for another limit.
2. Continuity of the function in Problem 2.
3. Derivative definition in Problem 3.
4. Tangent line equation in Problem 4.
5. Break-even analysis in Problem 5.
6. Differentiation in Problem 6.
7. Implicit differentiation in Problem 7.
8. Finding absolute extrema in Problem 8.

**Tip:** When facing the indeterminate form \( \frac{0}{0} \), techniques like factoring, rationalizing, or using L'Hôpital's Rule can help in finding the limit.

Learn how to solve a limit problem where x approaches 2. This problem involves rationalizing the denominator and applying limit rules to find the solution. Suitable for advanced high school students and college-level calculus enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation