An intuitive understanding of the limiting process

An Introdction to Limits
• Supose you are asked to graph the function f given by
$f(x)=\frac{x^3-1}{x-1}$

• 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 2.71 2.97 2.997 ? 3.003 3.03 3.31 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

$\lim_{x\to1}f(x)=3$
• 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
$\lim_{x\to(c)}f(x)=L$

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
$\lim_{x\to(c)}f(x)=L$ means that for each ε > 0 there exists a δ > 0 such that if 0 < |x-c| < δ, then
|f(x)-L| < ε.

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

example an intuitive understanding of the limit process