Question

find the limit.

$$\lim_{y \to -3}(5 - y)^{\frac{2}{3}}$$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. $$\lim_{y \to -3}(5 - y)^{\frac{2}{3}} = $$ (simplify your answer. type an integer or a simplified fraction.)

b. the limit does not exist.

Explanation:

Step1: Substitute \(y = - 3\) into the function

Substitute \(y=-3\) into \((5 - y)^{\frac{2}{3}}\). We get \(5-(-3)=5 + 3=8\). So the function becomes \(8^{\frac{2}{3}}\).

Step2: Simplify the exponent

We know that \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). For \(8^{\frac{2}{3}}\), we can write it as \(\sqrt[3]{8^{2}}\) or \((\sqrt[3]{8})^{2}\). Since \(\sqrt[3]{8}=2\) (because \(2\times2\times2 = 8\)), then \((\sqrt[3]{8})^{2}=2^{2}\).

Answer:

A. \(\lim_{y
ightarrow - 3}(5 - y)^{\frac{2}{3}}=4\)