Find the function values that are approximations for e....

Find the function values that are approximations for e. Round to five decimal places.
$$
\begin{aligned}
&\text { For } f(t)=(1+t)^{1 / t}, \text { we have } e=\lim _{t \rightarrow 0} f(t) . \text { Find } f(1)\\
&f(0.5), f(0.2), f(0.1), \text { and } f(0.001)
\end{aligned}
$$

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Key Concepts

Rounding

Rounding is the process of adjusting the digits of a number to reduce it to a simpler or more manageable form while keeping it close to its original value. In numerical approximations, rounding to a specified number of decimal places helps ensure consistency and clarity in reported results.

Function Evaluation and Approximation

Function evaluation involves substituting specific values into a function to compute approximate numerical results. This is particularly useful for approximating limits and understanding the behavior of a function near a point of interest when an exact expression is difficult to obtain.

Limit

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular value. They are used to analyze and understand trends in functions, particularly at points where the function may not be explicitly defined and form the basis for the definitions of derivatives and integrals.

Euler's Number (e)

Euler's number, often denoted as e, is a key mathematical constant approximately equal to 2.71828. It can be defined as the limit of expressions such as (1 + t)^(1/t) as t approaches zero, showcasing its role in continuously compounded growth and in the natural logarithm function.

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