In Exercises 17-22, sketch the graph of the function f and...

In Exercises 17-22, sketch the graph of the function $f$ and evaluate $\lim _{x \rightarrow a} f(x)$, if it exists, for the given value of $a$.
$f(x)=\left\{\begin{array}{ll}x-1 & \text { if } x \leq 0 \\ -1 & \text { if } x>0\end{array} \quad(a=0)\right.$

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Key Concepts

One-Sided Limits

One-sided limits are used to evaluate the behavior of a function as the input approaches a specific point from the left (negative side) or right (positive side). They are crucial for functions defined piecewise, as the function’s expression may change at the point of interest, affecting the limit from each side.

Graphical Representation of Functions

Graphing a function provides a visual interpretation of its behavior across its domain. This is especially useful for piecewise functions, as it helps illustrate points of discontinuity or changes in the function's formula, and confirms the results obtained through analytical evaluations such as limits.

Piecewise-Defined Functions

This concept involves functions that have different expressions over different parts of their domain. Understanding piecewise-defined functions requires examining each piece separately, particularly at the boundaries where the definition changes, to determine the overall behavior of the function.

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