https://file.aiquickdraw.com/mathbot/file/17273181007704qn0vqhr.jpg

The image presents a limit problem:

\[
\lim_{{x \to 8}} \frac{\frac{x}{x+1} - \frac{8}{9}}{x - 8}
\]

### Tasks:
1. A table of values for the function is created using \( x \) values approaching 8 from both sides (left and right). The values of the function \( f(x) \) are computed and rounded to four decimal places.
   
   The table of values provided is:
   - For \( x = 7.9 \), \( f(x) = 0.0125 \)
   - For \( x = 7.99 \), \( f(x) = 0.0124 \)
   - For \( x = 7.999 \), \( f(x) = 0.0123 \)
   - For \( x = 8.001 \), \( f(x) = 0.0123 \)
   - For \( x = 8.01 \), \( f(x) = 0.0123 \)
   - For \( x = 8.1 \), \( f(x) = 0.0122 \)

2. The table is then used to estimate the limit. The function values approach approximately \( 0.0123 \) as \( x \) gets closer to 8. Therefore, the estimated limit is:

\[
\lim_{{x \to 8}} \frac{\frac{x}{x+1} - \frac{8}{9}}{x - 8} \approx 0.0123
\]

Would you like more detailed explanations on how the function is derived or any clarifications on the steps?

### Five Related Questions:
1. What is the concept of limits in calculus?
2. How do you apply L'Hopital's Rule to solve limits involving indeterminate forms?
3. Can this limit be solved algebraically without creating a table?
4. How do you interpret the results of a limit approaching a value from both sides?
5. How can a graphing utility confirm the calculated limit?

### Tip:
When dealing with limits that result in indeterminate forms like \( \frac{0}{0} \), applying L'Hopital's Rule or algebraic simplification can often provide a direct solution.

Consider the following: lim(x→8) [(x/(x+1)) - (8/9)] / (x - 8). Create a table of values for the function and use the table to estimate the limit.

Learn how to estimate the limit lim(x→8) [(x/(x+1)) - (8/9)] / (x - 8) using a table of values. This problem applies concepts of limits and indeterminate forms with algebraic methods. Includes a step-by-step solution suitable for high school and college students.

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