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

Sketch a function that has the following table of values, but whose limit as $x \rightarrow 2$ does not exist:
$$
\begin{array}{|r|c|c|c|c|c|c|c|}
\hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\
\hline f(x) & 3.12 & 3.09 & 3.01 & - & 2.99 & 2.92 & 2.87
\end{array}
$$

Key Concepts

Limit of a Function

The limit of a function at a point describes the value that the outputs of the function approach as the input values get arbitrarily close to that point. It is determined by analyzing the behavior of the function from both sides of the point, and for a limit to exist, the values obtained from any sequence approaching that point must converge to the same number.

One-Sided Limits

One-sided limits focus on the behavior of a function as the input approaches a specified value exclusively from one direction—either from the left or the right. The overall limit at that point exists only when both the left-hand limit and the right-hand limit exist and are equal. Disparity between them causes the limit at the point to fail to exist.

Discontinuities

A discontinuity is a point at which a function is not continuous, meaning the function may have a break, jump, or oscillatory behavior near that point. In cases where the two one-sided limits do not agree or where the function exhibits erratic behavior, the overall limit does not exist. Discontinuities are key in understanding how a function can behave well away from a point while still lacking a limit at that point.

Piecewise Function Construction

Piecewise function construction involves defining different expressions for a function over different parts of its domain. This technique is useful for achieving specific functional behaviors—such as matching a prescribed set of values on one interval while deliberately engineering different behavior (like oscillation or jumps) near a certain point to cause the limit to not exist.

Please give Ace some feedback