Sketch the graph of the function and use it to determine the...

Sketch the graph of the function and use it to determine the values of $a$ for which $\lim _{x \rightarrow a} f(x)$ exists.
$$
f(x)=\left\{\begin{array}{ll}
\sqrt[3]{x} & \text { if } x \leqslant-1 \\
x & \text { if }-1<x \leqslant 2 \\
(x-1)^{2} & \text { if } x>2
\end{array}\right.
$$

Key Concepts

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a specific value exclusively from the left or right. For piecewise functions, it is important to examine these separately at the points where the function's definition changes to determine if the overall limit exists.

Continuity

Continuity involves checking whether a function has no breaks, jumps, or holes at a given point. For piecewise functions, ensuring continuity at the boundary points requires that the left-hand and right-hand limits are equal and match the function's value (if defined) at that point.

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain. This requires careful analysis at the boundaries where the definition changes, as the behavior of the function might differ when approaching these points from different sides.

Limits

The concept of a limit addresses the value that a function approaches as its input approaches a particular point. Limits are essential for understanding the behavior of functions near points of discontinuity or at boundaries of piecewise definitions.

Graphical Analysis

Graphical analysis entails visualizing the function by sketching its graph, which helps in understanding its behavior, identifying discontinuities, and verifying the existence of limits as the input approaches specified values. This tool is especially useful for piecewise functions where the behavior can change across different intervals.

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