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