if ( k(x)=\frac{x^{3}+512}{x + 8} ), complete the table and use the results to find ( lim _{x \rightarrow-8}…

Explanation:

Step 1: Substitute (x = - 8.1) into (k(x)=\frac{x^{3}+512}{x + 8})

[ \begin{align*} k(-8.1)&=\frac{(-8.1)^{3}+512}{-8.1 + 8}\ &=\frac{-531.441+512}{-0.1}\ &=\frac{-19.441}{-0.1}\ &=194.441 \end{align*} ]

Step 2: Substitute (x=-8.01) into (k(x)=\frac{x^{3}+512}{x + 8})

[ \begin{align*} k(-8.01)&=\frac{(-8.01)^{3}+512}{-8.01 + 8}\ &=\frac{-513.624001+512}{-0.01}\ &=\frac{-1.624001}{-0.01}\ &=162.400 \end{align*} ]

Step 3: Substitute (x = - 8.001) into (k(x)=\frac{x^{3}+512}{x + 8})

[ \begin{align*} k(-8.001)&=\frac{(-8.001)^{3}+512}{-8.001+8}\ &=\frac{-512.192007+512}{-0.001}\ &=\frac{-0.192007}{-0.001}\ &=192.007 \end{align*} ]

Step 4: Substitute (x=-7.999) into (k(x)=\frac{x^{3}+512}{x + 8})

[ \begin{align*} k(-7.999)&=\frac{(-7.999)^{3}+512}{-7.999 + 8}\ &=\frac{-511.808009+512}{0.001}\ &=\frac{0.191991}{0.001}\ &=191.991 \end{align*} ]

Step 5: Substitute (x=-7.99) into (k(x)=\frac{x^{3}+512}{x + 8})

[ \begin{align*} k(-7.99)&=\frac{(-7.99)^{3}+512}{-7.99+8}\ &=\frac{-510.407999+512}{0.01}\ &=\frac{1.592001}{0.01}\ &=159.200 \end{align*} ]

Step 6: Substitute (x=-7.9) into (k(x)=\frac{x^{3}+512}{x + 8})

[ \begin{align*} k(-7.9)&=\frac{(-7.9)^{3}+512}{-7.9 + 8}\ &=\frac{-493.039+512}{0.1}\ &=\frac{18.961}{0.1}\ &=189.610 \end{align*} ]

Answer:

(x)	(-8.1)	(-8.01)	(-8.001)	(-7.999)	(-7.99)	(-7.9)
(k(x))	(194.441)	(162.400)	(192.007)	(191.991)	(159.200)	(189.610)

As (x) approaches (-8) from both sides, (k(x)) approaches (192). So (\lim_{x\rightarrow - 8}k(x)=192)