Find the limit.

ModifyingBelow lim With y right arrow minus 2 left parenthesis 6 minus y right parenthesis Superscript four thirdslimy→−2(6−y)43




To find the limit \(\lim_{y \to -2} (6 - y)^{\frac{4}{3}}\), let's proceed with direct substitution since the expression inside the limit is continuous at \(y = -2\).

1. **Substitute \(y = -2\) directly:**

\[
6 - y = 6 - (-2) = 6 + 2 = 8
\]

2. **Evaluate the expression:**

\[
(6 - y)^{\frac{4}{3}} = 8^{\frac{4}{3}}
\]

3. **Simplify \(8^{\frac{4}{3}}\):**

\[
8^{\frac{4}{3}} = \left(2^3\right)^{\frac{4}{3}} = 2^{3 \cdot \frac{4}{3}} = 2^4 = 16
\]

So, the limit is:

\[
\lim_{y \to -2} (6 - y)^{\frac{4}{3}} = 16
\]

Would you like more details on any step or have any questions? Here are 5 related questions to consider:

1. How do you compute limits for more complex expressions involving radicals and exponents?
2. What happens when substituting into the limit yields an indeterminate form?
3. How do you compute limits involving infinity?
4. What are the properties of exponents that can simplify such limits?
5. How would the limit change if the exponent were negative?

**Tip:** Direct substitution is often the first step in evaluating limits for continuous functions. If you obtain an indeterminate form, consider algebraic simplifications or L'Hôpital's rule.

Learn how to find the limit of (6 - y)^4/3 as y approaches -2. Follow a step-by-step solution involving direct substitution and exponentiation. Suitable for advanced high school students and anyone studying calculus.

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