Warmup 1.3
Determine the following limits
1.
lim
2x 1
x 5 3 x  7
2.
lim
3.
lim
3 x
x 4 5 
x 3
4.
10  2 x
x  25 3 x  75
5.
lim cos x
6.
lim cos x
x 0
lim
7.
x 1
x 1
x 1
lim f  x 
x 2
b) lim f  x 
x 1
c) lim f  x 
x 1
d) lim f  x 
x 1
e) lim f  x 
x 1
lim
x
9. Given the graph of y  f  x  to the right,
determine
a)
x 3
8.
x2  5x  6
x2  9
π
2
lim  ln x 
x e
Limits Involving Infinity
Vertical asymptotes: Limits Equaling Infinity at finite values
If lim f ( x)    or lim f ( x)    then
x a
x a
For a rational function of the form f ( x) 
g ( x)
h( x)
, x  a is a vertical asymptote if
If both g  a  and h  a  equal zero, then there is either a hole in the graph or a break.
Find any vertical asymptotes. Describe the behavior to the left and right of any asymptotes.
1)
f ( x)  12
x
3)
f ( x) 
x  2
x2  4
2)
f ( x) 
1
x2  4
4)
f ( x) 
x2  4
x  2
5)
f ( x) 
x2  5x  6
x 2  8 x  12
6)
f ( x)  tan x
Finite limits as x   (Limits at the extreme left and right of the function)
k

xn
k
lim n 
x  x
lim
x 
Horizontal asymptotes – Limits at infinity
If lim f ( x)  b or lim f ( x)  b then
x 
x
Note: The maximum number of horizontal asymptotes that a function can have is _______. Not every
function has a horizontal asymptote. A function may have a horizontal asymptote on one side, but not
necessarily on the other side. A horizontal asymptote can be intersected by the function an infinite
number of times.
Determine any horizontal asymptotes. To do this, it means that you have to examine the behavior of the
function at both positive and negative infinity.
1)
f ( x) 
5x2  7 x
13x 2  12
3)
f ( x) 
2 x4  5 x
4 x3  1
4)
f ( x)  sin x
5)
f ( x) 
cos x
x
6)
f  x   ex
7)
f ( x) 
5sin x
x  8x
8) y  ln x
2
2) y 
x3  3 x  1
4 x3  7
It is sometimes helpful when exploring limits at infinity to use an end behavior model. An end behavior
model is a simple, basic function which closely approximates the actual function at either infinity or
negative infinity.
For the function f:
The function g is a right end behavior model if lim
x 
The function g is a left end behavior model if lim
x 
f  x
1
g  x
f  x
1
g  x
For the following functions, determine right and left end behavior models:
Function
y  4 x5  7 x3  3x2  8
y  3x  x 2
y  2 x3  5 x
y  6 x  log x
y  sin  1x 
y  9 x 2  2053
y
1
3
x
y
4 x3  5 x  2
12  3x 2  7 x
y
18 x
36 x 2  10000 x
Left end behavior model
Right end behavior model
Determine all asymptotes for the function f ( x) 
4x
x  4
2
“Seeing” limits at infinity – To explore limits at infinity on a graphics calculator, the question is how far
do you have to go to see what is happening at infinity. One way to explore this graphically is to examine
1
the graph of y  f   and examine the behavior of this transformed function around zero.
x
1
1
This means lim f  x   lim f   and lim f  x   lim f  
x 
x

x 0
x

0
 x
 x