Limits and Asymptotes Insert
Properties of Limits
If
lim f ( x)  L
x c
and
lim g ( x)  M , then
x c
Constant Rule
lim k  k
xc
Identity Rule
lim x  c
x c
Sum/Difference Rule
lim ( f ( x)  g ( x))  L  M
x c
Constant Multiple Rule
lim (k  f ( x))  k  L
x c
Product Rule
lim ( f ( x)  g ( x))  L  M
x c
Quotient Rule
lim
x c
f ( x) L

; M 0
g ( x) M
Power Rule
If r and s are integers, s  0 , then
lim ( f ( x)) r / s  Lr / s
x c
r/s
provided that L is a real number.
Using the rules above, we can also apply the limit rules to polynomial and rational functions.
***You saw these rules last year.
sin 𝑥
𝑥→0 𝑥
One limit that you must know is lim
= 0. We can then use that limit to find others, such as
sin( 2 x)
tan x
sin( 3 x)
lim
lim
,
, and x0
.
x 0
x 0
sin( 3x)
x
x
lim
The Sandwich Theorem
Infinity is an unbounded quantity that is higher than any real number. (definition taken from: Weisstein, Eric W.
"Infinity." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Infinity.html )
1
on your calculator.
x
1
1
 0.
Notice that as x gets bigger,
gets closer and closer to zero, also written as lim
x


x
x
Ex 1) Graph the line y 
1
0
Similarly, xlim
 x
1
actually be equal zero?” The answer is NO, but we could make it
x
1
 0!!!
arbitrarily close to zero. That’s why lim
x x
Now the real question is, “can
y=0 is called a Horizontal Asymptote of the graph y 
1
.
x
Horizontal Asymptote
Ex 2) Investigate the graph of y
x
.
x 1
2
Does it have a horizontal asymptote?
Side note: The sandwich theorem also holds for limits as x  
Finding a limit as x approaches infinity
sin(
x)
Ex 3) Find lim
.
x
x
When we have a limit as x approaches infinity the top and the bottom typically have a “race to infinity.”
If the numerator wins the race the limit of the function is infinity. If the denominator wins, the function
approaches zero. A shortcut to this is by looking for the fastest growing term.
The order is as follows:
1.
2.
3.
4.
Constants grow at the slowest rate (a constant rate)
Logs
Polynomials
Exponentials grow at the fastest rate
23y2
lim
Ex 4) Determine y5y2 4y
Ex 5) Determine lim
x 
2x
x2
1ex
Ex 6) Determine lim
x12ex
Limits at infinity have the same properties as the properties of finite limits that we talked about on the
first page of this document.
Practice: Calculate the horizontal and vertical asymptotes for the functions below.