12.1 Limits

If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L.

Note: As x gets closer to c, the closer y gets to L

Example:

• One way to solve this is by putting 2 in for x.
• Another way is to make a table

x 1.9 1.99 1.999 2 2.001 F(x) 4.7 4.97 4.997 5 5.003

The closer x gets to 2 (from both sides) the closer f(x) gets to 5

Describing the limit

Limits that Do Not Exist (DNR):

1. when the left and right behaviors are different

the limit from the left would be -2, but the limit from the right would be 2.

The limit from the left side is described as

The - sign above the 2 shows it is from the left side

The limit from the right side is 2 (c) with a + sign

2. If a function increases without a bound, f(x) can be as large as you want

The asymptote of x=0 causes you to be able to choose a number as close to 0 and it keeps decreasing

3. When it Oscillates between two different numbers

the limit just switches between 1 and -1.

Continuity:

If a function is continuous at c then

• A trick for knowing if a graph is continuous is if you can draw it out without lifting up your pencil.
• Discontinuous graphs include: BREAKS, HOLES AND ASYMPTOTES