lim
x
→
3

3
√
9
x
−
3
√
3
+
x
−
√
2
x
=

To solve the limit 

\[
\lim_{x \to 3} \left( \sqrt[3]{9x} - \sqrt[3]{3 + x} - \sqrt{2x} \right)
\]

we'll evaluate each term separately to determine if we can apply direct substitution or if further techniques are required.

### Step 1: Substitute \( x = 3 \) Directly
Let's attempt direct substitution to see if it results in a defined value.

1. **First Term:** \( \sqrt[3]{9x} = \sqrt[3]{9 \times 3} = \sqrt[3]{27} = 3 \)
2. **Second Term:** \( \sqrt[3]{3 + x} = \sqrt[3]{3 + 3} = \sqrt[3]{6} \)
3. **Third Term:** \( \sqrt{2x} = \sqrt{2 \times 3} = \sqrt{6} \)

Putting these values into the limit expression, we get:

\[
3 - \sqrt[3]{6} - \sqrt{6}
\]

### Conclusion
Since each term evaluated individually gives a finite value, the limit is:

\[
\lim_{x \to 3} \left( \sqrt[3]{9x} - \sqrt[3]{3 + x} - \sqrt{2x} \right) = 3 - \sqrt[3]{6} - \sqrt{6}
\]

Would you like more details, or do you have any questions?

---

### Related Questions to Explore
1. How would this limit change if \( x \to 0 \) instead of \( x \to 3 \)?
2. What techniques could help if we encountered an indeterminate form like \( \frac{0}{0} \)?
3. How do we evaluate limits involving higher roots, like the fourth root or fifth root?
4. What are some applications of limits in real-world scenarios?
5. How can we use L'Hôpital's Rule when there are radical expressions?

**Tip:** When substituting directly into a limit, always check if it leads to a defined number. If not, try algebraic manipulation or L'Hôpital's Rule.

lim x → 3 (3√(9x) − 3√(3 + x) + x − √(2x)) =

Discover how to solve the limit lim x → 3 (3√(9x) − 3√(3 + x) − √(2x)) using direct substitution. This solution explores the calculation of cube and square roots within limit problems, providing a step-by-step approach. Ideal for students in advanced high school calculus.

Math Problem Statement

3 √ 9 x − 3 √ 3 + x − √ 2 x

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation