https://file.aiquickdraw.com/mathbot/file/1726694236379dww8gmf8.jpg

The problem asks you to estimate the limit of the function 

\[
f(x) = \left(1 - \frac{1}{x}\right)^x
\]

as \( x \to \infty \), both using a graph (to two decimal places) and using a table (to three decimal places).

### Part a) Graph Estimation

When \( x \) becomes very large, the function \( f(x) \) converges towards a specific value. The function

\[
\lim_{x \to \infty} \left(1 - \frac{1}{x}\right)^x
\]

is a well-known limit that approaches the mathematical constant \( e^{-1} \), or approximately 0.367879. For the purpose of graphing, we truncate this value to two decimal places, which gives **0.37**.

### Part b) Table Estimation

Similarly, using a table where \( x \) is allowed to increase, we can observe that the limit approaches \( e^{-1} \), which is approximately 0.367879. Truncated to three decimal places, the answer would be **0.367**.

#### Answers:
- Part a): 0.37
- Part b): 0.367

Would you like more details or have any questions?

### Relative Questions:
1. What is the general behavior of functions as \( x \to \infty \)?
2. How do you graphically estimate a limit using a calculator or graphing tool?
3. Why is the limit of this function equal to \( e^{-1} \)?
4. How can you use a table of values to estimate a limit to more precision?
5. What other common limits involve the constant \( e \)?

### Tip:
When working with limits at infinity, it's helpful to recognize known forms or patterns that relate to exponential growth or decay.

Estimate lim(x -> infinity) (1 - 1/x)^x using a graph truncated to two decimal places and a table truncated to three decimal places.

Learn how to estimate the limit of (1 - 1/x)^x as x approaches infinity using graphing tools and a table. The problem covers important concepts like limits, exponential functions, and asymptotic behavior, with step-by-step solutions. Suitable for Grades 11-12 or introductory calculus students.

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