Let's analyze the problem in the image.

Problem Statement:

You are asked to estimate the limit $\lim_{{x \to a}} g(x)$ or determine if it does not exist. The function given is:

$g(x) = \frac{{e^{9x} - 9x - 1}}{{x^2}} \quad \text{at} \quad a = 0$

First Task:

• Graphical Estimation of the Limit: You need to select the correct graph to estimate the limit or conclude that it does not exist.

Second Task:

• Numerical Evaluation: The table shows values of $g(x)$ for values of $x$ approaching 0 from both sides (positive and negative), and you need to use these values to support your conjecture about the limit.

Observations:

1. You are given three graph options, each representing different behaviors of the function near $x = 0$.
2. The table has values for $g(x)$ at different $x$ near 0, with values like $0.1, 0.01, -0.01,$ etc., and you need to compare these to estimate the limit.

Solution Steps:

1. Graph Analysis: Based on the shapes of the graphs provided, you are looking for the one where the function approaches a finite value or continues to grow significantly, implying the limit exists or does not.

2. Numerical Table Analysis: Analyze the table of values to see how $g(x)$ behaves as $x$ approaches 0. Does the function settle around a particular number, or does it become unbounded (goes to infinity)?

3. Limit Evaluation:

   • If the function values approach a specific number, the limit exists.
   • If the values diverge (go to infinity), the limit does not exist.

Do you have any questions about this problem or want further clarifications?

Here are five questions to expand your understanding:

1. How can you analytically find limits involving $\frac{{e^x - 1}}{{x}}$ type functions?
2. What does it mean if a limit does not exist at a point?
3. Can L'Hopital's Rule be applied here for finding this limit analytically?
4. What would happen to the limit if $e^{9x}$ were replaced with a different exponential function?
5. What real-world phenomena could this type of limit model?

Tip: When dealing with limits of exponential functions near 0, remember that the derivative of $e^x$ at 0 is 1, which can simplify the analysis.