*Limits* logically come before continuity since the definition of continuity requires using limits. But practically and historically, continuity comes first. The concept of a limit is used to explain the various kinds of discontinuities and asymptotes. Start by studying discontinuities.

Types of discontinuities to consider: removable (a gap or hole in the graph), jump, infinite (vertical asymptotes), oscillating, and end behavior (horizontal asymptotes).

Numerically: Make a table for the value of near *x* = 3 and as . Relate the values and their signs to the graph. (Divide by a small number get a big number; divide by a big number, get a small number.)

Use the vocabulary of limits to explain the features of graphs. Example: The function has no value at *x* = 2 (*f*(2) does not exist), but as you get closer to *x* = 2 the function value gets closer to 4 ().

Relate the limit, value and graph of the function. In the example above, the graph *looks like* the line with a gap or hole at the point (2, 4). Another example: since, the graph gets closer to y = 3 as you go farther to the right. The line *y* = 3 is a horizontal asymptote.

Do this numerically as well: and since the fraction gets smaller as |*x*| gets larger, the function approaches 3 from above when *x* > 0 and from below when *x* < 0 (why?)

Extra for your experts: Discuss the reason for the jump discontinuity of

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