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

Find the function values that are approximations for e. Round to five decimal places.
$$
\begin{aligned}
&\text { For } g(t)=t^{1 /(t-1)}, \text { we have } e=\lim _{t \rightarrow 1} g(t) . \text { Find } g(0.5)\\
&g(0.9), g(0.99), g(0.999), \text { and } g(0.9998)
\end{aligned}
$$

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Key Concepts

Limit of a Function

This concept involves studying the behavior of a function as its input approaches a particular value. In mathematical analysis, limits are used to define continuity, derivatives, and pivotal constants by examining how functions behave near points of interest, especially when direct substitution leads to indeterminate forms.

Natural Exponential Constant (e)

The constant e is a fundamental mathematical constant that arises in various branches of mathematics, particularly in calculus and exponential growth or decay processes. It is defined as the limit of certain sequences or functions, such as the one given in this problem, serving as the base of natural logarithms and exponential functions.

Numerical Approximation and Rounding

In many practical situations, exact analytical solutions may not be directly computable, so numerical methods are employed to approximate values. Rounding, especially to a specified number of decimal places, is an important technique used to express these approximations with a desired precision, ensuring the results are both usable and accurate.

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