BC 1
Limits 5
We have shown the following:
Name:
lim
x0
1  cos x
sin x
 0.
=1 and lim
x 0
x
x
Now, find each of the following limits. Guess and check your answer with graphs and
tables on your calculator.

1  cos 3 x
sin 6x
lim
lim
(1)
(2)
(3)
lim
 0 sin 4
x 0
x0
x
x
(4)
lim
x0
sin 6x
sin 2x
(5)
lim
x0
tan 3x
2x
For each of the following, justify your work. (Use correct notation.)
tan 3x
sin 3 x 1
1
sin 3 x 3 x
3 3
 lim

 lim


 11 
Example: (5) above: lim
x 0
x 0 cos 3 x 2 x
x 0 cos 3 x
2x
3x 2 x
2 2
(6)
lim
(7)
lim
(8)
lim
x 0
sin 5x
2x
sin 2 2x
x0
x2
 0
tan 4
sin 8
(9) Use a graphing utility to guess lim
0
sinh( )
of the graph of f ( x)  sinh( x) at x  0?
IMSA BC 1

. What does this tell you about the slope
Lim 5.1
x 3
. Find all points of discontinuity. Determine whether each is a
| x | 3
removable discontinuity or not.
(10)
Let f ( x) 
(11)
Write the equation of a function with vertical asymptotes at x = 2 and x = 5, a
removable discontinuity at x =  1 and a horizontal asymptote of y = 2. Check
your function by graphing.
(12)
Find lim
(13)
Let
(14)
Let f (x ) 
x 
x + sin x
.
x + cos x
4 x  5, x  1
f ( x)  
. Is ƒ continuous at x =  1? Justify.
2
 3  x , x  1
tan x
tan x
. It is possible to find lim
by methods used above.
x
x0
x
tan x
Instead, consider the inequality 1 <
< 1 + x2 if x  0, x near 0.
x
Convince yourself that the inequalities seem Use the Squeeze Theorem to find the limit
reasonable by using your calculator to sketch above.
the three functions on the same graph.
IMSA BC 1
Lim 5.2
p
1
 if x  is a rational # in (0,1) reduced to lowest terms
q
(15) Let f ( x)   q
0
if x is a irrational # in (0,1)

Sketch the graph of y  f ( x) below:
Do you think that lim f ( x) exists for a  (0,1) ? Explain. You may want to discuss the
xa
cases of a rational and a irrational separately.
IMSA BC 1
Lim 5.3