Supose you are asked to graph the function f given by

At any point other than x=1 standard mathematical techniques will give you the solution. So to get some idea of how the function would behave near x=1, use two sets of x values one set below 1 and one set above 1.

x

0.75

0.9

0.99

0.999

1

1.001

1.01

1.1

1.25

f(x)

2.313

2.710

2.970

2.997

?

3.003

3.030

3.310

3.813

Since the parabola f has a hole at the point (1.3) x¹1, but you can move very close to 1 and because of that you can move very close to 3

This gives you the basic understanding of what a Limit is and what role it preforms.

A formal Definition of Limit
*So the informal definition of a Limit is as f(x) becomes very close to a number (L) as x approaches c from either side, which means the limit of f(x) as x aproaches c is L. Shown as

The formal mathematical definition of a limit is as follows:

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement

means that for each ε > 0 there exists a δ > 0 such that if 0 < |x-c| < δ, then
|f(x)-L| < ε.

*ε represents a small positive number *δ represents a positive number
Larson, Hosteler, Edward, Calculus Eighth Edition

An Introdction to Limitsfgiven byf(x)A formal Definition of Limit*So the informal definition of a Limit is as f(x) becomes very close to a number (L) as x approaches c from either side, which means the limit of f(x) as x aproaches c is L. Shown as

The formal mathematical definition of a limit is as follows:

- Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement

means that for each ε > 0 there exists a δ > 0 such that if 0 < |x-c| < δ, then|f(x)-L| < ε.

*ε represents a small positive number

*δ represents a positive number

Larson, Hosteler, Edward, Calculus Eighth Edition

example an intuitive understanding of the limit process