Questions: If k(x)=(x^3+512)/(x+8), complete the table and use the results to find lim x→−8 k(x). x -8.1 -8.01 -8.001 -7.999 -7.99 -7.9 x Complete the table. x -8.1 -8.01 -8.001 -7.999 -7.99 -7.9 k(x) (Round to three decimal places as needed.)

If k(x)=(x^3+512)/(x+8), complete the table and use the results to find lim x→−8 k(x). x -8.1 -8.01 -8.001 -7.999 -7.99 -7.9 x Complete the table. x -8.1 -8.01 -8.001 -7.999 -7.99 -7.9 k(x) (Round to three decimal places as needed.)
Transcript text: If $k(x)=\frac{x^{3}+512}{x+8}$, complete the table and use the results to find $\lim _{x \rightarrow-8} k(x)$. \begin{tabular}{l|llllll} $\mathbf{x}$ & -8.1 & -8.01 & -8.001 & -7.999 & -7.99 & -7.9 \\ \cline { 1 - 1 }$(\mathbf{x})$ & & & & & & \end{tabular} Complete the table. \[ \begin{array}{c|cccccc} \mathbf{x} & -8.1 & -8.01 & -8.001 & -7.999 & -7.99 & -7.9 \\ \mathbf{k}(\mathbf{x}) & \square & \square & \square & \square & \square & \square \end{array} \] (Round to three decimal places as needed.)
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Solution

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Solution Steps

Step 1: Evaluate \( k(x) \) at Given Points

We evaluate the function \( k(x) = \frac{x^{3} + 512}{x + 8} \) at the specified values of \( x \):

  • For \( x = -8.1 \), \( k(-8.1) \approx 194.41 \)
  • For \( x = -8.01 \), \( k(-8.01) \approx 192.24 \)
  • For \( x = -8.001 \), \( k(-8.001) \approx 192.024 \)
  • For \( x = -7.999 \), \( k(-7.999) \approx 191.976 \)
  • For \( x = -7.99 \), \( k(-7.99) \approx 191.76 \)
  • For \( x = -7.9 \), \( k(-7.9) \approx 189.61 \)
Step 2: Fill in the Table

The values of \( k(x) \) corresponding to the \( x \) values are as follows: \[ \begin{array}{c|cccccc} \mathbf{x} & -8.1 & -8.01 & -8.001 & -7.999 & -7.99 & -7.9 \\ \mathbf{k}(\mathbf{x}) & 194.41 & 192.24 & 192.024 & 191.976 & 191.76 & 189.61 \end{array} \]

Step 3: Estimate the Limit

To estimate \( \lim_{x \rightarrow -8} k(x) \), we calculate the average of the \( k(x) \) values as \( x \) approaches -8: \[ \text{Estimated limit} \approx \frac{194.41 + 192.24 + 192.024 + 191.976 + 191.76 + 189.61}{6} \approx 192.003 \]

Final Answer

The limit as \( x \) approaches -8 is approximately \( \boxed{192.003} \).

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