Sketch a function that has the following table of values,...

Sketch a function that has the following table of values, but whose limit as $x \rightarrow \infty$ is equal to $-\infty$ :
$$
\begin{array}{|r|c|c|c|c|c|}
\hline x & 100 & 200 & 500 & 1,000 & 10,000 \\
\hline f(x) & 50 & 55 & 56.2 & 56.89 & 56.99
\end{array}
$$

Key Concepts

Piecewise Defined Functions

Piecewise functions are constructed by defining different rules for different parts of the domain. In the context of the problem, even though the function takes on near-constant values at certain inputs, one can design the function in pieces so that it agrees with the table at specified points while being defined differently elsewhere to ensure the limit as x tends to infinity is -?. This concept is essential when trying to create a function that meets both discrete value constraints and specified asymptotic behavior.

Interpolation and Function Construction

Interpolation involves designing a function that passes through a given set of points. The problem shows a table of values that the function must match at specific inputs, and the concept of function construction here emphasizes that while the function must pass through those points, it also has significant freedom in how it behaves between or beyond these points. This freedom can be exploited to ensure that the overall limit of the function as x approaches infinity meets the required condition.

Limit at Infinity

This concept refers to the behavior of a function as its input grows without bound. When we say that the limit of a function as x approaches infinity is -?, we mean that for sufficiently large x, the function decreases without bound, even if it has local or isolated behavior that seems different. This is crucial in reconciling a behavior that appears nearly constant at certain sample points with the overall trend of the function at large values of x.

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