When you work with limit and continuity problems in calculus, there are a couple of formal definitions you need to know about. So, before you take on the following practice problems, you should first re-familiarize yourself with these definitions.

Here is the formal, three-part *definition of a limit*:

For a function *f* (*x*) and a real number *a*,

exists if and only if

(Note that this definition does not apply to limits as *x* approaches infinity or negative infinity.)

Now, here's the *definition of continuity*:

A function *f* (*x*) is continuous at a point *a* if three conditions are satisfied:

Now it's time for some practice problems.

## Practice questions

Using the definitions and this figure, answer the following questions.

At which of the following

*x*values are all three requirements for the existence of a limit satisfied, and what is the limit at those*x*values?*x*= –2, 0, 2, 4, 5, 6, 8, 10, and 11.At which of the

*x*values are all three requirements for continuity satisfied?

## Answers and explanations

All three requirements for the existence of a limit are satisfied at the

*x*values 0, 4, 8, and 10:At 0, the limit is 2.

At 4, the limit is 5.

At 8, the limit is 3.

At 10, the limit is 5.

To make a long story short, a limit exists at a particular

*x*value of a curve when the curve is*heading toward*some particular*y*value and keeps*heading toward*that*y*value as you continue to zoom in on the curve at the*x*value. The curve must head toward that*y*value (that height) as you move along the curve both from the right and from the left (unless the limit is one where*x*approaches infinity).The phrase

*heading toward*is emphasized here because what happens precisely at the given*x*value isn't relevant to this limit inquiry. That's why there is a limit at a hole like the ones at*x*= 8 and*x*= 10.The function in the figure is continuous at 0 and 4.

The common-sense way of thinking about continuity is that a curve is continuous wherever you can draw the curve without taking your pen off the paper. It should be obvious that that's true at 0 and 4, but not at any of the other listed

*x*values.