Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Precalculus and ACT - Calculating Limits Using the Limit Laws | MATH 1910, Study notes of Mathematics

Material Type: Notes; Professor: Lamb; Class: MATH 1910: If high school precalculus and ACT math of at least 26 contact 694-6450.; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 08/19/2009

koofers-user-1t8
koofers-user-1t8 🇺🇸

10 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
MATH 1910
Section 2.3 Calculating Limits Using the Limit Laws
Look at the limit laws beginning on page 111 of the textbook. An important note is that the limit laws can only
be applied to functions whose limits EXIST.
Ex. Given that
lim ( )
xa
fx
=2 lim ( )
xa
gx
=
4 lim ( )
xa
hx
=
1,
evaluate the following limits, if they exist.
a) lim[ () ()]
xa
fx hx
3 b) lim[ ( )]
xa
gx
2 c) lim ( )
()
()
xa
gx
hx fx
+
d)
lim ()
xa
fx
1 e)
lim ( ) ( )
xa
hx gx
3
3
Ex. Evaluate the following limits, if they exist, without the aid of a calculator:
a)
lim
x
x
x
x
→−
+−
12
2
43
b) lim
x
xx
→−
++
1
327
There are times when you have to do some algebraic maneuvering such as factoring both sides of a rational
expression, rationalizing the NUMERATOR of a rational expression before you can evaluate a limit.
Ex. Evaluate the following limits, if they exist, without the aid of a calculator:
a) lim
x
xx
x
→−
−−
+
3
212
3 b) lim
x
x
x
1
2
3
1
1 c) lim
h
h
h
+−
0
55
=−+
+
→−
lim ()()
x
xx
x
3
43
3 =
lim ()()
()( )
x
xx
xxx
+
−++
12
11
11
=
lim
h
h
h
h
h
+− ++
++
0
5555
55
cancel out the ( )
x
+3 = lim ()
()
x
x
xx
+
++
12
1
1 rationalize the numerator
=−
→−
lim ( )
x
x
3
4= – 7 = 2
3 =
lim ()
h
h
hh
+−
⋅++
0
55
55
=
lim ()
h
h
hh
⋅++
055
=
lim
hh
++
=
0
1
55
1
25
Ex. Evaluate the one sided limit lim
xxx
+
L
N
MO
Q
P
0
11
using your knowledge of the absolute value function.
pf2

Partial preview of the text

Download Precalculus and ACT - Calculating Limits Using the Limit Laws | MATH 1910 and more Study notes Mathematics in PDF only on Docsity!

MATH 1910

Section 2.3 Calculating Limits Using the Limit Laws

Look at the limit laws beginning on page 111 of the textbook. An important note is that the limit laws can only be applied to functions whose limits EXIST.

Ex. Given that lim ( ) x a f x → = 2 lim ( ) x a g x → = − 4 lim ( ) x a h x

evaluate the following limits, if they exist.

a) lim[ ( ) ( )] x a f x h x → 3 − b) lim[ ( )] x a g x

(^2) c) lim (^ ) ( ) x a (^ )

g xh x^ + f^ x

d) lim (^) ( ) xa f^ x

(^1) e) lim ( ) ( ) x a h x g x

Ex. Evaluate the following limits, if they exist, without the aid of a calculator:

a) lim x

x →− x x

b) lim x x x →−

1

There are times when you have to do some algebraic maneuvering such as factoring both sides of a rational expression, rationalizing the NUMERATOR of a rational expression before you can evaluate a limit.

Ex. Evaluate the following limits, if they exist, without the aid of a calculator:

a) lim x

x x →− x

b) lim x

xx

2 3

c) lim h

hh

0

= −^ +

lim (^ )(^ ) x

x x (^3) x

= lim (^ )(^ ) x ( )( )

x xx x x

= lim h

h h

hh

cancel out the ( x + 3 ) = lim (^ ) x ( )

xx x

rationalize the numerator

= − →− lim ( ) x x 3

= lim h ( )

hh h

= lim h ( )

h → (^0) h ⋅ 5 + h + 5 = lim h → (^) + h +

0

Ex. Evaluate the one sided limit lim x → +^ x x

L

N

M

O

Q

0 P

(^1 1) using your knowledge of the absolute value function.

MATH 1910

Section 2.3 Calculating Limits Using the Limit Laws

Ex. For the function f x

e x x x x x

x ( )

R

S

T

. Which of the following limits exist:

a) lim ( ) x f x → 0 b) lim ( ) x f x → 1 c) lim ( ) x f x → 2 HINT : Try not to graph the function, just use the y-values on the different intervals and your knowledge of the functions involved.

THE SQUEEZING THEOREM : If f ( ) xg x ( ) ≤ h x ( )when x is near a (except possible at a) and lim ( ) lim ( ) x a x a f x h x L → → = = then lim ( ) x a g x L

Ex. Use the Squeeze Theorem to show that lim sin x x x

0

3 d i^1 .

First, notice we can’t use the limit laws and split the limit up as lim lim sin x x x x → →

0 0

3 d i^1 because lim sin

x → 0 x

d i^1 does

not exist. But, we do know one thing about the values of sin d i^1 x , due to the nature of the trig function,

− 1 ≤ sin d i^1 x ≤ 1.

How can we now apply the squeeze theorem?