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I. The Limit Laws, Summaries of Calculus

The limit of a constant function is equal to the constant. 8 ax ax. = → lim. The limit of a linear function is equal to the number x is approaching.

Typology: Summaries

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Math131 Calculus I The Limit Laws Notes 2.3
I. The Limit Laws
Assumptions: c is a constant and )(lim xf
ax and )(lim xg
axexist
Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f,
then =
)(lim xf
ax
“Simpler Function Property”: If )()( xgxf
=
when
a
x
then )(lim)(lim xgxf axax
=, as long as the
limit exists.
Limit Law in symbols Limit Law in words
1 )(lim)(lim)]()([lim xgxfxgxf axaxax
+=+ The limit of a sum is equal to
the sum of the limits.
2 )(lim)(lim)]()([lim xgxfxgxf axaxax
= The limit of a difference is equal to
the difference of the limits.
3 )(lim)(lim xfcxcf axax
= The limit of a constant times a function is equal
to the constant times the limit of the function.
4 )](lim)(lim)]()([lim xgxfxgxf axaxax
= The limit of a product is equal to
the product of the limits.
5 )(lim
)(lim
)(
)(
lim xg
xf
xg
xf
ax
ax
ax
=
(
)
0)(lim
xgif ax
The limit of a quotient is equal to
the quotient of the limits.
6 n
ax
n
ax xfxf )](lim[)]([lim
= where n is a positive integer
7 cc
ax
=
lim The limit of a constant function is equal
to the constant.
8 ax
ax
=
lim The limit of a linear function is equal
to the number x is approaching.
9 nn
ax ax =
lim where n is a positive integer
10
nn
ax ax =
lim where n is a positive integer & if n is even,
we assume that a > 0
11
n
ax
n
ax xfxf )(lim)(lim
= where n is a positive integer & if n is even,
we assume that )(lim xf
ax
> 0
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

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Math131 Calculus I The Limit Laws Notes 2.

I. The Limit Laws

Assumptions: c is a constant and lim^ f ( x )

xa

and lim^ g ( x )

xa exist

Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f ,

then lim x → a^ f ( x^ )=

“Simpler Function Property”: If f ( x )= g ( x )when x ≠ a then lim f ( x ) lim g ( x )

xa xa

= , as long as the

limit exists.

Limit Law in symbols Limit Law in words

1 lim x → a [^ f^ ( x )+^ g ( x )]=lim x → af ( x )^ +lim x → ag ( x ) The limit of a sum is equal to

the sum of the limits.

2 lim x → a [^ f^ ( x )−^ g ( x )]=lim x → af ( x )^ −lim x → ag ( x )

The limit of a difference is equal to the difference of the limits.

3 lim x → a^ cf^ ( x )^ =^ c lim x → af ( x ) The limit of a constant times a function is equal

to the constant times the limit of the function.

4 lim x → a [^ f^ ( x ) g ( x )]=^ lim x → af ( x )^ ⋅lim x → ag ( x )]

The limit of a product is equal to the product of the limits.

lim ( )

lim ( )

lim

g x

f x

g x

f x

x a

x a x a

→ →

= ( lim ( )≠ 0 )

if (^) xagx

The limit of a quotient is equal to the quotient of the limits.

n x a

n x a

lim[ f ( x )] [lim f ( x )]

→ →

= where n is a positive integer

7 lim x → a^ c^ = c

The limit of a constant function is equal to the constant.

8 lim x → a^ x^ = a The limit of a linear function is equal

to the number x is approaching.

n n x a

x = a

lim where n is a positive integer

10 lim x → a^ nx^ = n^ a where^ n^ is a positive integer & if^ n^ is even,

we assume that a > 0

11 lim x → a^ n f^ ( x )^ =^ n lim x → af (^ x )

where n is a positive integer & if n is even, we assume that (^) lim f ( x ) xa

Math131 Calculus I Notes 2.3 page 2

ex#1 Given (^) lim ( ) 2 3 = → f x x

, (^) lim ( ) 1 x → 3 gx =−

, (^) lim ( ) 3 3 = → hx x

use the Limit Laws find (^) lim () ( )^2 ( ) 3 f xhx xg x x − →

ex#2 Evaluate 6 4

2 1 lim (^2)

2 (^2) + −

→ (^) x x

x x , if it exists, by using the Limit Laws.

ex#3 Evaluate: (^) lim 2 2 3 5 x → 1 x +^ x^ −

ex#4 Evaluate: x

x x

2 0

lim

ex#5 Evaluate: h

h h

lim 0

Math131 Calculus I Notes 2.3 page 4

ex#11 What is lim[[ ]]

3

x

x

Theorem 2: If f ( x )≤ g ( x )when x is near a (except possibly at a ) and the limits of f and g both

exist as x approaches a then lim f ( x ) lim g ( x ) xa xa

ex12 Find x

x x

lim 2 sin → 0

. To find this limit, let’s start by graphing it. Use your graphing calculator.

The Squeeze Theorem: If f ( x )≤ g ( x )≤ h ( x )when x is near a (except possibly at a ) and

f x hx L x a x a

→ →

lim ( ) lim ( ) then g x L x a

lim ( )

Math131 Calculus I Limits at Infinity & Horizontal Asymptotes Notes 2.

Definitions of Limits at Large Numbers

Theorem

  • If r > 0 is a rational number then 0

lim = x → ∞ (^) xr

  • If r > 0 is a rational number such that x r is defined for all x then 0

lim = x → −∞ xr

Definition in Words Precise Mathematical Definition

Large POSITIVE numbers

Let^ f^ be a function defined on some interval ( a ,^ ∞). Then f x L x

→ ∞ lim ( ) means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large in a positive direction.

Let f be a function defined on some interval ( a , ∞). Then f x L x

→ ∞

lim ( ) if for every ε > 0 there is a

corresponding number N such that if x > N then

f ( x )− L < ε

Large NEGATIVE^ numbers

Let^ f^ be a function defined on some interval (-∞, a ). ∞). Then f x L x

→ −∞ lim ( ) means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large in a negative direction.

Let f be a function defined on some interval (-∞, a ). Then f x L x

→ −∞

lim ( ) if for every ε > 0 there

is a corresponding number N such that if x < N then

f ( x )− L < ε

Definition What this can look like…

Horizontal Asymptote

The line y = L is a horizontal asymptote of the curve y = f(x) if either is true:

  1. f x L x

→ ∞ lim ( ) or

  1. f x L x

→ −∞ lim ( )

Vertical Asymptote

The line x = a is a vertical asymptote of the curve y = f(x) if at least one of the following is true:

  1. =∞ → lim f ( x ) x a
  2. (^) − =∞ →

lim f ( x ) x a

  1. (^) + =∞ →

lim f ( x ) x a

  1. =−∞ → lim f ( x ) x a
  2. (^) − =−∞ →

lim f ( x ) x a

  1. (^) + =−∞ →

lim f ( x ) x a

Math131 Calculus I Notes 2.6 page 3

ex#5 Find the vertical and horizontal asymptotes of the graph of the function: 3 5

2

x

x f x

"or*

C.

, (^) !!$rP$L 1. o+{-i y^ - g r4-,+-CR.\ r:

  • orot^ a\r.eA^ -\ob \o\k-i^ ^g $e.^ dov^ FroA..cf oq^ -

o".. (^) c.

;l+' .o' (^) o+ e^<-

a-,.- (^) - o..- ""q+-r x^ A^ €o<

i+6 co\v*^ (^) o{

< \ a_j^ s,p

a

. --,o,

I P^@^ J4''

\X ^{

r1\i 6\io.i!iov\ |^ +* ..t .A"

(^1) A'(\5+ Q) = (^) A.5 + AC. +* (^) AeM^-S) 6.c€l{--(e.

= (^) lA'IA tL, ) -< !4-e

\rts (^) sh^(/Js

oc <ea!-