11-12 Sketch the graph of the function and use it to...

11-12 Sketch the graph of the function and use it to determine the values of a for which $\lim _{x \rightarrow a} f(x)$ exists. $$f(x)=\left\{\begin{array}{ll}{1+x} & {\text { if } x<-1} \\ {x^{2}} & {\text { if }-1 \leq x<1} \\ {2-x} & {\text { if } x \geqslant 1}\end{array}\right.$$

11-12 Sketch the graph of the function and use it to determine the values of a for which $\lim _{x \rightarrow a} f(x)$ exists.
$$f(x)=\left\{\begin{array}{ll}{1+x} & {\text { if } x<-1} \\ {x^{2}} & {\text { if }-1 \leq x<1} \\ {2-x} & {\text { if } x \geqslant 1}\end{array}\right.$$

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Key Concepts

Graphical Analysis

Graphical analysis involves sketching or visualizing the function to identify points of discontinuity, jumps, and trends. This is particularly useful for piecewise functions, as it provides a visual representation of how the different segments of the function connect or fail to connect at transition points, which in turn reveals the existence or non-existence of limits at those points.

One-sided Limits

One-sided limits refer to the values that a function approaches as x approaches a specific point from either the left (left-hand limit) or the right (right-hand limit). For the overall limit to exist at a point, the left-hand and right-hand limits must both exist and be equal.

Limits of Functions

The limit of a function as x approaches a particular value involves understanding the behavior of the function arbitrarily close to that point. The limit exists if the values of the function approach a specific number regardless of the direction from which x approaches the target point.

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain. Understanding how each piece is defined and where the conditions change is crucial for analyzing its behavior overall, especially at the boundary points where the formula changes.

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