









Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
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
1 / 17
This page cannot be seen from the preview
Don't miss anything!
Math131 Calculus I The Limit Laws Notes 2.
x → a
x → a exist
Direct Substitution Property: If f is a polynomial or rational function and a is in the domain of f ,
x → a x → a
limit exists.
Limit Law in symbols Limit Law in words
the sum of the limits.
The limit of a difference is equal to the difference of the limits.
to the constant times the limit of the function.
The limit of a product is equal to the product of the limits.
x a
x a x a →
→ →
if (^) x → agx
The limit of a quotient is equal to the quotient of the limits.
n x a
n x a
→ →
The limit of a constant function is equal to the constant.
to the number x is approaching.
n n x a
→
we assume that a > 0
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
3
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
lim = x → ∞ (^) xr
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
→ ∞
corresponding number N such that if x > N then
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
→ −∞
is a corresponding number N such that if x < N then
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:
→ ∞ lim ( ) or
→ −∞ 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
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*
, (^) !!$rP$L 1. o+{-i y^ - g r4-,+-CR.\ r:
o".. (^) c.
;l+' .o' (^) o+ e^<-
a-,.- (^) - o..- ""q+-r x^ A^ €o<
i+6 co\v*^ (^) o{
< \ a_j^ s,p
a
\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!-