https://file.aiquickdraw.com/mathbot/file/1726518389530ujcd2h84.jpg

The limit presented in the image is:

\[
\lim_{x \to 8} \left( 2x^{2/3} - 7x^{-1} \right)
\]

Let's solve this step-by-step.

### Step 1: Plug in \(x = 8\)
For limits, the first approach is usually to substitute the value directly if the function is well-behaved at that point.

#### Evaluate \( 2x^{2/3} \) at \(x = 8\):
\[
2 \times 8^{2/3}
\]
\(8^{2/3}\) is the cube root of 8 squared. Since \( 8 = 2^3 \), we get:
\[
8^{2/3} = (2^3)^{2/3} = 2^2 = 4
\]
Thus,
\[
2 \times 4 = 8
\]

#### Evaluate \( -7x^{-1} \) at \(x = 8\):
\[
-7 \times 8^{-1} = -7 \times \frac{1}{8} = -\frac{7}{8}
\]

### Step 2: Combine the results:
\[
\lim_{x \to 8} \left( 2x^{2/3} - 7x^{-1} \right) = 8 - \frac{7}{8} = \frac{64}{8} - \frac{7}{8} = \frac{57}{8}
\]

### Final Answer:
\[
\frac{57}{8}
\]

Would you like more details or further clarification?

### Related Questions:
1. What is the concept of cube roots and fractional exponents?
2. How does direct substitution work in evaluating limits?
3. What are some special cases where direct substitution doesn't work for limits?
4. Can limits be evaluated using L'Hôpital's rule in certain indeterminate forms?
5. How can you graph functions to visualize their behavior near certain points?

**Tip:** For functions involving fractional exponents, understanding basic root properties simplifies limit evaluations significantly.

Find the limit: lim (x → 8) [2x^(2/3) - 7x^(-1)]

Learn how to solve the limit problem lim (x → 8) [2x^(2/3) - 7x^(-1)] using direct substitution. This guide covers limits, fractional exponents, and substitution methods. Suitable for students in Grades 10-12.

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