https://file.aiquickdraw.com/mathbot/file/1726048262830ovye1ogw.jpg

To solve the limit for \( x \to 1^+ \) in problem 61, let's analyze the given expression:

\[
\lim_{x \to 1^+} \left( \frac{1}{x^{2/3}} + \frac{2}{(x - 1)^{2/3}} \right)
\]

### Step 1: Consider the behavior of each term as \( x \) approaches 1 from the right (\( x \to 1^+ \)).

1. **Term 1: \( \frac{1}{x^{2/3}} \)**
   - As \( x \) approaches 1, \( x^{2/3} \) approaches \( 1^{2/3} = 1 \).
   - Therefore, \( \frac{1}{x^{2/3}} \) approaches \( \frac{1}{1} = 1 \).

2. **Term 2: \( \frac{2}{(x - 1)^{2/3}} \)**
   - As \( x \) approaches 1 from the right, \( x - 1 \) becomes a small positive number.
   - Consequently, \( (x - 1)^{2/3} \) becomes a small positive number raised to the power \( 2/3 \), which makes \( \frac{2}{(x - 1)^{2/3}} \) a very large positive number because the denominator is approaching 0 from the positive side.

### Step 2: Combine the terms
- The first term \( \frac{1}{x^{2/3}} \) is approximately 1 as \( x \to 1^+ \).
- The second term \( \frac{2}{(x - 1)^{2/3}} \) dominates and approaches \( +\infty \) as \( x \to 1^+ \).

### Conclusion:
The limit is dominated by the second term, which tends to \( +\infty \). Therefore, the limit is:

\[
\lim_{x \to 1^+} \left( \frac{1}{x^{2/3}} + \frac{2}{(x - 1)^{2/3}} \right) = +\infty
\]

---

Would you like more details on this solution or do you have any other questions?

### Related Questions:
1. What is the limit of \( \lim_{x \to 0^+} \left(\frac{1}{x^{2/3}}\right) \)?
2. How does \( (x - 1)^{2/3} \) behave as \( x \) approaches 1 from the left?
3. Can we compute a similar limit for \( x \) approaching 0 instead of 1?
4. What happens if \( x \) approaches 2 instead of 1 in the given expression?
5. How does the behavior change if the exponent \( 2/3 \) is modified?

### Tip:
When dealing with limits involving terms of the form \( (x - a)^{n} \) where \( n \) is a fraction, pay special attention to whether \( x \) approaches \( a \) from the left or the right. The behavior can differ significantly based on the sign of the term inside the parentheses.

Solve 61 (c): lim_{x -> 1^+} (1/x^{2/3} + 2/(x - 1)^{2/3})

Solve the limit problem lim_{x -> 1^+} (1/x^{2/3} + 2/(x - 1)^{2/3}) with a step-by-step explanation. This problem explores key calculus concepts involving limits and fractional exponents, suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation