An Introdction to Limits
  • 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.

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

example an intuitive understanding of the limit process