Sketch graphs by hand and use them to make approximations...

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits. $$ \lim _{x \rightarrow-1} f(x), \text { for } f(x)=\left\{\begin{aligned} x+1, & \text { if } x<-1 \\ 2, & \text { if } x=-1 \\ -x^{2}, & \text { if } x>-1 \end{aligned}\right. $$

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow-1} f(x), \text { for } f(x)=\left\{\begin{aligned}
x+1, & \text { if } x<-1 \\
2, & \text { if } x=-1 \\
-x^{2}, & \text { if } x>-1
\end{aligned}\right.
$$

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Key Concepts

Graphical Analysis in Calculus

Graphical analysis involves sketching or interpreting the graphs of functions to visually approximate their behavior near specific points. This approach is valuable for understanding limits, particularly with piecewise functions or functions with potential discontinuities.

Piecewise Functions

Piecewise functions are defined using different expressions over different intervals of their domain. When evaluating limits for these functions, it is important to examine the behavior of each segment of the function around the point of interest to determine overall limit behavior.

Limits

A limit describes the value that a function approaches as the input approaches a particular point, even if the function is not defined exactly at that point. This concept is fundamental in calculus as it forms the basis for defining derivatives and continuity.

One-Sided Limits

One-sided limits consider the value a function approaches as the input approaches a specific point from only one side—either from the left (x ? a?) or from the right (x ? a?). This is key for functions that exhibit different behaviors on either side of a point, helping to analyze discontinuities.

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