Limits and Continuity

Limit and Continuity


Definition: Limit at a real number c (Both side limit).

Given > 0, there exists > 0 such that | f ( x ) - L | < whenever 0 < | x - c | < ,
then the limit of f ( x ) at x = c is L, and denoted by ,
otherwise f ( x ) has no limit.

Roughly speaking, means that whenever x approaches to c

from either side of c, the graph point (x, f(x)) approaches to the point (c, L)

on the plane.


Definition: Continuity for f ( x ) at x = c ( both side continuity ).

Given > 0, there exists > 0, such that | f ( x ) - f ( c ) | < whenever | x - c| < ,
then f ( x ) is continuous at x = c. ( i.e ),
otherwise f ( x ) is not continuous at x = c. ( i.e f ( x ) is discontinuous at x = c ).

Roughly speaking, means that whenever x approaches

to c from either side of c, the graph point (x, f(x)) approaches to the

point (c, f(c)) on the plane.


Remarks:

( 1 ) The existence of the limit for f ( x ) at x = c is not related to the existence

of the output of f ( x ) at x = c.

( 2 ) When , f ( x ) is continuous at x = c.
( 3 ) Whenever does not exist or ,
then f ( x ) is not continuous at x = c.


exists but is not equal to f ( c ), then the graph of f ( x ) has a hole at x = c.
does not exist, then the graph of f ( x ) has either a jump at x = c or
f ( x ) has a vertical asymptote at x = c ( will be defined later ).


Example1: Show that
Analysis :

( 1 ) Given > 0, we need to select > 0 to satisfy that | f ( x ) - L | < whenever 0 < | x - 4 | < .

( 2 ) Since ( x - 4 ) = for x > 0, we need to show < from the condition

0 < | x - 4 | < ( i.e ) ( * ), the factor should be combined with .

By rearranging ( * ), we acquire . But, we need . We can achieve this by

restricting the x values close to 4 such that the lower bound of can be obtained.

Therefore, we can obtained a relationship between and . e.g. let ( i.e ),

then . So, i.e. .