1.5 Limits
Graphically, Numerically, and Analytically
Informal definition of a limit:
If
becomes arbitrarily close to a single finite number L as x
approaches c from both sides then it can be stated that the limit of
x approaches c is L. This is written mathematically as
as
lim f ( x)  L .
x c
Limits that fail to exist.
1. Unbounded behavior.
2. f (x) approaches two different values as x approaches c from either
side.
Finding Limits Analytically
Try direct substitution.
1. If the result is finite, then the limit is L.
2. If the result is
nonzero
, the limit does not exist.
zero
0
3. If the limit is indeterminate, that is
, try to use the “replacement
0
theorem”
If f ( x)  g ( x) for all x in an interval containing c, except possibly at
x  c , and if the limit of g(x) exists, and is equal to L, then
lim g ( x)  lim f ( x)  L .
x c
x c
“How to find g(x) “.
 Factor and cancel.
 Rationalize and cancel.
 Simplify and cancel.
 Expand and cancel.
One Sided Limits
A One Sided Limit:
lim f ( x)
x c
lim f ( x)
x c
, the limit of f (x) as x approaches c from the left.
, the limit of f (x) as x approaches c from the right.
Theorem: The limit of f (x) as x approaches c from both sides equals L if and
only if the limit of f (x) as x approaches c from the left equals L and limit of
f (x) as x approaches c from the right equals L.
lim f ( x)  L
xc
Section 1.6
lim f ( x)  lim f ( x)  L
if and only if x c 
Continuity
Removable vs. non-removable discontinutiy
Examples of Discontinuity
x c
.
Continuity at a Point and on an Open Interval
End Point Discontinuity