12.1 Intro to Limits

Definition of Limit

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.

Official Notation:

Simplified Definition:

As x get closer to a point from both side, what would the y or f(x) value most likely be without considering what it actually is.

The limit is what the approaching numbers imply, NOT the actual value.

Solve by Graphing:

Function  

Limit

By looking at the graph, as x approaches the value 1, the line gets closer to the y value 6.

Answer

Types of limits

A limit can either exist or not exist with a graph. Existing one may look a lot like the one above with a few varieties. Non-existing limits can be separated into two categories, one that don't exist and infinities.

A limit as x approaches 2 wouldn't exist.
Either side of the limit approach different numbers.

A limit at 1 could be two things.
(depending on strictness of definition)
the answer could be infinity or does not exist.

With limits that have not definite infinity, they lye on the line between existence. The definition says that it is a number, and since infinity is not a single number, it doesn't exist. But infinity is also true and is useful in calculus.

Solve with a Table:

This process involves plugging in number extremely close to the limit. If both sides of the limit get extremely close to a number, that number is the limit.