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. $$ \lim _{x \rightarrow 1} f(x), \text { for } f(x)=\left\{\begin{aligned} x^{2}+1, & \text { if } x<1 \\ 3, & \text { if } x=1 \\ 3-x, & \text { if } x>1 \end{aligned}\right. $$

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.
$$
\lim _{x \rightarrow 1} f(x), \text { for } f(x)=\left\{\begin{aligned}
x^{2}+1, & \text { if } x<1 \\
3, & \text { if } x=1 \\
3-x, & \text { if } x>1
\end{aligned}\right.
$$

Key Concepts

Graphical Analysis of Limits

Graphical analysis is a visual method to approximate and understand the behavior of a function near a point. By sketching the graph, one can observe the trends from both the left and right sides, making it easier to approximate the limits and identify any discontinuities or abrupt changes in the function’s behavior.

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a specific value from one side – either from the left (denoted as the left-hand limit) or from the right (denoted as the right-hand limit). They are particularly useful in analyzing functions with different definitions on either side of a point and in identifying discontinuities such as jumps in the graph.

Limit of a Function

A limit describes the behavior of a function as the input approaches a particular value. It is a fundamental concept in calculus that helps in understanding how a function behaves near a point, regardless of the function's value at that point. This concept is essential for analyzing the continuity, differentiability, and overall behavior of functions.

Piecewise Functions

Piecewise functions are defined by different expressions depending on the input value's interval or region. This concept is important because each piece of the function may behave differently, and understanding the limits, continuity, or differentiability of the overall function necessitates examining each piece individually.

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