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.