§2 Limits and continuity 2.1 Some important definitions and results about functions

Note: Note:

2.2 Limit of a function Given a function f(x), if x approaching 3 causes the function to take values approaching (or equaling) some particular number, such as 10, then we will call 10 the limit of the function and write In practice, the two simplest ways we can approach 3 are from the left or from the right.

For example, the numbers 2.9, 2.99, 2.999,... approach 3 from the left, which we denote by x→3 –, and the numbers 3.1, 3.01, 3.001,... approach 3 from the right, denoted by x→3 +. Such limits are called one-sided limits.

Example: We make two tables, as shown below, one with x approaching 3 from the left, and the other with x approaching 3 from the right.

2.3 One-Sided Limit We have introduced the idea of one-sided limits. We write

We write

2.4 Methods of finding One-Sided Limit Left-hand Limit: c c-h h

2.4.2 Right-hand Limit: c h

2.5 The precise ( ) definition of Limit a L

Some important results

The Definition of Limit a L

Examples What do we do with the x?

Limit Theorems

The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. a L One-Sided Limits

The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. a M

Theorem