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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 )
x → a
and lim^ g ( x )
x → a 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 )
x → a x → a
= , 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 (^) x → agx
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 ) x → a
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 ) x → a x → a
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:
- f x L x
→ ∞ lim ( ) or
- 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:
- =∞ → lim f ( x ) x a
- (^) − =∞ →
lim f ( x ) x a
- (^) + =∞ →
lim f ( x ) x a
- =−∞ → lim f ( x ) x a
- (^) − =−∞ →
lim f ( x ) x a
- (^) + =−∞ →
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!-
