Limits

Intuition#

When gets closer and closer to a number as approaches , we call the limit of at . The dashed line in the figure is that .

The precise definition#

Limit

Let be defined on a punctured neighbourhood of . If for every there is a such that implies

then is the limit of at , written .

The value at the point does not matter

The definition only looks at . Whether is defined, and what it equals, has no effect on the limit.

An example#

Prove that

Given , choose . Whenever ,

Limit laws#

If and , then

With limits in hand, Derivatives can be defined.