Introduction to LimitsDoes 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;
ExamplesEvaluate the following limits and tell whether
the limit exist.
lim 2x+3x ⟶ 1
lim 2x+3x ⟶ 0
lim 2x+3x ⟶ -3/2
= 2(1)+3=5, ∴ limit exists
= 2(0)+3=3, ∴ limit exists
= 2(-3/2)+3=0, ∴ limit exists
When does the limit of a function not exist?
lim 2x+3x ⟶ -3 x+3
= -30
∴ limit does not exist
Properties of limitsConstant
Scalar Multiple
Sum or difference
Product
Quotient
Power
Techniques for Evaluating LIMITS
FISH
Dividing Out Technique
lim x-6x ⟶ 6x2-36
(6)-6
(36)-36
0
0undefined
Factor
lim x-6x ⟶ 6
(X-6)(x+6)
1
(x+6)
1
(6+6)
1
12∴ limit exists
Practice:
lim 5-xx ⟶ 515-3x
lim x2-81x ⟶ 9 9-x
lim x2-12x+36x ⟶ 1x2-7x+6
=⅓ , ∴ limit exists
=-18 , ∴ limit exists
= -5/0 , ∴ limit DOES NOT exist
