Download BC 1 Limits 5 Name: We have shown the following: lim sinx x

BC 1
Limits 5
We have shown the following:
Name:
lim
x!0
sin x
1" cos x
= 1 and lim
= 0.
x!0
x
x
Now, find each of the following limits. Guess and check your answer with graphs and
tables on your calculator.
sin6x
!
1" cos3x
(1)
(2)
(3)
lim
lim
lim
x!0
!
"
0
x!0
x
sin4!
x
(4)
lim
x!0
sin6x
sin2x
(5)
lim
x!0
tan 3x
2x
For each of the following, justify your work. (Use correct notation.)
tan 3x
sin3x 1
1
sin3x 3x
3 3
Example: (5) above: lim
= lim
"
= lim
"
"
= 1"1" =
x!0
x!0
x!0
2x
cos3x 2x
cos3x 3x 2x
2 2
(6)
lim
(7)
lim
(8)
lim
(9)
x !0
sin 5x
2x
sin2 2x
x!0
x2
!" 0
tan 4!
sin 8!
Find lim
IMSA BC 1
!" 0
sin!
when θ is in degrees. Explain why this result makes sense.
!
Lim 5.1
Spr 06
x!3
. Find all points of discontinuity. For each, determine whether
| x | !3
or not it is a removable discontinuity.
(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)
lim
Find x!"
(13)
Let
(14)
Let f (x) =
x + sin x
.
x + cos x
# 4x + 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!0
x
x
tan x
2
Instead, consider the inequality 1 <
< 1 + x 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
Spr 06