1.3 Evaluating Limits Analytically

Objectives:-Students will evaluate a limit using properties of limits-Students will develop and use a strategy for finding limits-Students will evaluate a limit using dividing out and rationalizing techniques-Students will evaluate a limit using the Squeeze Theorem

Properties of Limits

let lim ( ) and lim ( )

scalar: lim[ ( )]

sum/diff: lim[ ( ) ( )]

product: lim[ ( ) ( )]

( )quotient: lim

g(x)

power: lim[ ( )]

direct

x c x c

x c

x c

x c

x c

n n

x c

f x L g x K

b f x bL

f x g x L K

f x g x LK

f x L

K

f x L

sub: lim ( ) ( ) if f is constant at cx cf x f c

Ex 1) let lim ( ) 2 and lim ( ) 3

a) lim[5 ( )]

b) lim[ ( ) ( )]

c) lim[ ( ) ( )]

( )d) lim

g(x)

x c x c

x c

x c

x c

x c

f x g x

g x

f x g x

f x g x

f x

Ex 2) Yesterday we found that

but we had to do this graphically; direct substitution didn’t work because of the hole (the denom was 0)

To find the limit analytically…simplify! Find the same function but without the hole.

3

1

1lim 3

1x

x

x

3

1

1lim

1x

x

x

x

y

Strategy for finding :

1) Direct substitution2) Simplify to an identical function

except at x=c ; then use direct substitution.

3) Use a graph or table to check.4) Remember, sometimes the limit

DNE!!

lim ( )x c

f x

Simplifying Techniques:

1) Factoring (cancellation)2

3

6lim

3x

x x

x

Simplifying Techniques:

2) Rationalization (rationalize numerator by multiplying by conjugate over itself)

0

1 1limx

x

x

Simplifying Techniques:

3) ∆x problems → expand then reduce

2 2

0

( )limx

x x x

x

Simplifying Techniques:

4) Common denominator

0

4limx

xx

x

Other Things to Look For:

Squeeze Theorem-if for all x in an open interval containing c, except possibly at c, and if , then

( ) ( ) ( )h x f x g x

lim ( ) lim ( )x c x ch x L g x

lim ( )

x cf x L

Two Special Trigonometric Limits:

** memorize these!!

0

sinlim 1x

x

x

0

1 coslim 0x

x

x

Ex 3) 0

tanlimx

x

x

Ex 4)

→ check using graphs or tables!

0

sin 4limx

x

x

Ex 5) 0

sec 1lim

sec