finding limits graphically
and numerically (1.2)
September 19th, 2012
I. An Introduction to Limits
Informal Definition: The limit of f(x), as x approaches c
from either side is the number L that f(x) becomes
f (x)  L .
arbitrarily close to. It is written lim
xc
x 2  3x  2
lim
x2
x2
Ex. 1: Estimate
numerically by completing
the table. Then use a graphing utility to estimate the limit
graphically.
x
f(x)
1.75
1.9
1.99 1.999
2
2.001 2.01
2.1
2.25
II. Limits That Fail to Exist
A. Behavior That Differs from the Right and Left
Ex. 2: Show that the limit does not exist.
x2
lim
x2 x  2
B. Unbounded Behavior
Ex. 3: Discuss the existence of the limit.
x
lim 2
x4 x  3x  4
C. Oscillating Behavior
Ex. 4: Discuss the existence of the limit.
 5
lim  sin  
x0
 x
Summary of Common Behavior Associated with the
Nonexistence of a Limit:
1) f(x) approaches a different number from the right side of
c than it approaches from the left side.
2) f(x) increases or decreases without bound as x
approaches c.
3) f(x) oscillates between two fixed values as x approaches
c.