https://file.aiquickdraw.com/mathbot/file/1726334441014bt6q5c65.jpg

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.

Use a graph of g(x) to estimate lim (g(x)) as x -> a, or show that the limit does not exist. Evaluate g(x) near x = a to support your conjecture. The function is g(x) = (e^(9x) - 9x - 1) / x^2 at a = 0. Use the graphing utility to estimate the limit or conclude that it does not exist.

Explore how to estimate the limit of the function (e^(9x) - 9x - 1) / x^2 as x approaches 0. Learn to evaluate this limit using graphical tools and numerical data, suitable for university-level calculus students. The solution also explains how L'Hopital's Rule could be applied for analytical solutions.

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