Introduction to Limits
Does limits of function exist?
Description of limits
Limits come in all different shapes
and sizes
Some functions have limits at all
points, while others contain
holes
Some functions may not have a
limit at all
Basic Limit Structure
x⟶c
The above limit is read as, “The limit of the function
f (x) as x approaches c.”
Basically, this means that as x approaches some
constant, f (x) will eventually become something;
Examples
Evaluate the following limits and tell whether the limit
exist.
lim 2x+3
x⟶1
= 2(1)+3 =5, ∴ limit exists
lim 2x+3
= 2(0)+3 =3, ∴ limit exists
lim 2x+3
= 2(-3/2)+3 =0, ∴ limit exists
x⟶0
x ⟶ -3/2
When does the limit of a
function not exist?
lim 2x+3
x ⟶ -3
x+3
=
-3
0
∴ limit does not exist
Properties of limits
Constant
Scalar Multiple
Sum or difference
Product
Quotient
Power
Techniques for
Evaluating LIMITS
Dividing Out Technique
lim x-6
x⟶6 2
x -36
lim x-6
x⟶6
(X-6)(x+6)
1
(6)-6
(36)-36
(x+6)
1
(6+6)
0
0
undefined
1
12
∴ limit exists
Practice:
lim 5-x
x⟶5
15-3x
lim
x⟶9
lim
x ⟶1
2
x -81
=⅓ , ∴ limit exists
=-18 , ∴ limit exists
9-x
2
x -12x+36
2
x -7x+6
= -5/0 , ∴ limit
DOES NOT exist