It looks like you’ve uploaded a problem that involves estimating limits numerically. The table includes values for $x$ and corresponding values of $f(x)$ for each of the following limits. I'll walk you through the steps for solving the exercises shown in the image.

Exercise 7: $\lim_{x \to 2} (5x + 4)$

You can estimate the limit by plugging in values of $x$ close to 2 and observing the behavior of $f(x)$. Let's fill in the table for $x = 1.9, 1.99, 1.999, 2, 2.001, 2.1$.

Solution:

• For each $x$, substitute into the function $f(x) = 5x + 4$ and compute the corresponding values of $f(x)$.
• As $x$ approaches 2, the function should approach $5(2) + 4 = 14$.

Exercise 8: $\lim_{x \to 1} (2x^2 + x - 4)$

Similarly, estimate the limit for values close to $x = 1$.

Solution:

• Plug in $x = 0.9, 0.99, 0.999, 1.001, 1.1$ into the function $f(x) = 2x^2 + x - 4$.
• As $x$ approaches 1, compute the corresponding values of $f(x)$.

Exercise 9: $\lim_{x \to 1} \frac{x + 1}{x^2 - x - 2}$

This is a rational function, so as $x$ approaches 1, observe the behavior near this value.

Solution:

• Substitute $x = -1.1, -1.01, -1.001, -1, -0.999$ into the function $f(x) = \frac{x + 1}{x^2 - x - 2}$.
• Observe the values of $f(x)$ to estimate the limit.

Exercise 10: $\lim_{x \to 0} \frac{\sin(2x)}{x}$

This involves a sine function, and it's typically useful for understanding limits involving trigonometric functions.

Solution:

• Substitute $x = -0.1, -0.01, -0.001, 0, 0.001$ into $f(x) = \frac{\sin(2x)}{x}$.
• As $x$ approaches 0, estimate the limit.

Would you like me to compute the exact values for these limits or provide a more detailed breakdown for one of the exercises? Let me know if you'd like further assistance with any of these!

Here are some related questions you could explore:

1. How does the behavior of $f(x)$ change as we get closer to the value for each exercise?
2. How can you recognize when a limit does not exist numerically?
3. What happens if you directly substitute $x = 2$ into the expression for exercise 7?
4. Can limits involving rational functions have undefined values? How do you handle such cases?
5. How does the small-angle approximation for sine relate to limits like exercise 10?

Tip: Always check if the function is continuous around the point of interest, as this can help predict the behavior of the limit.