Question

find the limit. (limlimits_{y \to -6} (2 - y)^{\frac{7}{3}})

Explanation:

Step1: Substitute \( y = -6 \) into the function

We know that for a continuous function \( f(y) \), \( \lim_{y \to a} f(y)=f(a) \). The function \( f(y)=(2 - y)^{\frac{7}{3}} \) is a composition of polynomial and power functions, which is continuous in its domain. So we substitute \( y=-6 \) into \( 2 - y \).
\( 2-(-6)=2 + 6=8 \)

Step2: Calculate the power of the result

Now we need to calculate \( 8^{\frac{7}{3}} \). We know that \( 8 = 2^{3} \), so we can rewrite \( 8^{\frac{7}{3}} \) as \( (2^{3})^{\frac{7}{3}} \).
Using the exponent rule \( (a^{m})^{n}=a^{mn} \), we have \( (2^{3})^{\frac{7}{3}}=2^{3\times\frac{7}{3}}=2^{7} \)
And \( 2^{7}=128 \)

Answer:

\( 128 \)