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1.02
INTRODUCTION TO LIMITS
DEFINITION OF A LIMIT
A limit is the intended value of a function.
It is the value which f(x) gets close to as "x"
gets close to "a".
SOMETIMES this limit is just f(a) but NOT
always.
DEFINITION OF A LIMIT

Speed limit  the speed which you can reach
but not go over

“I’ve hit my limit”  I’ve had enough, I can’t
take any more

In calculus, a limit is the intended value of a
function
EXAMPLE 1
6
y
4
2
x
−6
−4
−2
2
−2
−4
−6
4
6
lim  1
x 1
EXAMPLE 1
6
y
4
2
x
−6
−4
−2
2
−2
−4
−6
4
6
lim  6
x2
EXAMPLE 1
6
y
4
2
x
−6
−4
−2
2
−2
−4
−6
4
6
lim  2
x 0
EXAMPLE 2
lim  2
x4
EXAMPLE 3

A limit will not exist if the function is
approaching an undefined value (ie ∞ )
100
y
lim  undefined
x 3
50
x
−4
−2
2
4
6
8
RIGHT AND LEFT HAND LIMITS

lim
x 3

lim
x 3
means the limit approaching 3 from
the right
means the limit approaching 3 from
the left
lim  1
x  3
lim  3
x 3
RIGHT AND LEFT HAND LIMITS

For a limit to exist, the right-hand limit (RHL)
and the left-hand limit (LHL) must both exist
and must be equal
lim  1
x  3
lim 
x 3
Therefore, the
limit does not
3 exist
lim  DNE
x 3
EVALUATING LIMITS
 Limits
can be evaluated 3 ways:
Graphically
Algebraically
(several different method)
Using the Sandwich Theorem (only
some limits) also known as the squeeze
theorem
EVALUATING GRAPHICALLY WITH CALCULATOR

Use your graphing calculator to evaluate each
of the following limits (calculator should be in
RADIANS)
sin x
lim 
x 0
x
cos x  1
lim 
x 0
x
x
1
lim  (1  )
x
x 0
HOMEWORK

From the Finney textbook
 P. 62 # 1 – 6
 P. 64 # 45 – 47 (instructions are on p. 63)