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

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

Key Concepts

Graphical Analysis of Limits

Graphical analysis involves sketching or interpreting the graph of a function to visually identify how the function behaves near certain points. This method assists in approximating limits and understanding the existence or non-existence of two-sided limits, especially in scenarios involving discontinuities or split behaviors.

Limits

Limits describe the value that a function approaches as the input approaches a particular point. They are fundamental to calculus and help in understanding the behavior of functions near points where they may not be explicitly defined or may change behavior abruptly.

One-sided Limits

One-sided limits consider the behavior of a function as the input approaches a specific value from one direction – either from the left or from the right. This concept is particularly useful when functions exhibit different behavior on either side of a point, often leading to jump discontinuities.

Piecewise-Defined Functions

Piecewise-defined functions are functions that have different expressions based on the input value’s interval. Analyzing limits for such functions requires investigating the behavior separately on the different intervals and then comparing the limits as the value of interest is approached from each side.

Discontinuity

Discontinuity refers to points at which a function exhibits a break or a jump; the function does not have a single limit value as the input approaches that point. Understanding the type of discontinuity is crucial, particularly when dealing with piecewise-defined functions where the left-hand limit and right-hand limit may differ.

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