One activity is to define the number of regular sides that a shape must have in order to approximate a circle. For example, a square touches four points in a circle but is not very close to the actual shape. A pentagon meets the circle at five points but does not approximate the shape closely either. A regular shape with thousands of sides would be closer to the shape of the circle but still not identical because it does not have the elliptical shape, so the only method to solve the problem is by using a limit function.
Another important activity that can be done with limits is to examine basic graphs. For example, a rounded line with all positive numbers can be described as a limit as x approaches infinity. For a rounded line in the opposite quadrant on the graph, the equation would be the same except the line would approach negative infinity.
Another way of approaching a limit is to examine the graph of a straight line. For example, a line where x = 2 or y = 4 still approaches infinity. However, the x or y points will always be known on those particular lines, which makes these lines easier to work with. Students could be asked to draw these lines up until a certain point as a practice activity.
Limits can also be examined with inequalities. A line that is disjointed with two lines may use an inequality to describe its graph. For example, if a line is based on an inequality, then the limit as the line approaches a point will be different based on the direction from which it is coming. Ask students to graph these inequalities based on the limit function.