intro to limits/SOLUTIONS AT BOTTOM
A. Evaluate each of the following limits. Write DNE if the limit does not exist .
x 3
x 3
x
1. lim
2. lim
h 0
h
h
2/3
3. lim cos( )
 1
4. lim
h0
h 2  3h
h3
t2  7t  8
5. lim
t 1
1  t2
6. lim
 0
cos( )
sin( )
7. lim e 3x
x4
4
below:
a5
f(t  h)  f(t)
lim
h0
h
13. lim
t3
T (t)  T(1)
where T(t) =
t1
t 1
14. lim
x  3a
x  3a
t 1
a 2
15. lim sin(
ta
t
)
2a
16. lim
x 2  16
x 2  8x  16
17. lim
x  3a
x  3a
18. lim
h4  3h3
h3
x4
x 2
h0
19. lim ln(x)
x0
8. Use f(a) =
20. lim ln(cosx)
9. Use g(x) = 3 - x2 below:
21. lim ln(sin x)
lim
h0
g(3  h)  g(3)
h
10. lim ln( x)
x1
11. lim
t 0
12. lim
t0
x0
x0
22. lim
xa
x  3a
x  3a
4
1  et
23. lim tan x
4
1  et
24. lim csc 
x


2

2
B. Let H(t) =
sin(t),
t < /2
cos(t),
t > /2
Use this function to answer the following:
1. Evaluate: lim  H(t)
t  / 2
2. Evaluate:
3. Does
lim  H(t)
t  / 2
lim H(t) exist? If yes, what is the value of the limit? If no,
t / 2
explain why the limit does not exist. Sketch a graph of y = H(t).
C. Sketch the graph of a function y = K(t) that is continuous everywhere
except at t = -2 but lim K(t) exists.
t  2
D. Can you draw a function y = f(x) continuous on the open interval (0, 2)
which does not have an absolute maximum or an absolute minimum? Can you
draw a function y = g(x) continuous on the closed interval [0, 2] which does
not have an absolute maximum or an absolute minimum?
solutions to "intro to limits"
A. 1. 0
2.
1
lim h3  0
h0
3. cos() = -1
h3
DNE ( In a bit we'll describe the way the limit doesn't exist
h0
h2
by saying the limit is . )
4. lim
5. -
9
2
6. DNE 7. e12
4
4

4h
4
8. lim t  h  5 t  5 = lim
=
h0
h  0 h(t  h  5)(t  5)
h
(t  5)2
9. lim
h0
13. lim
t1
14.
6h  h2
3  (3  h)2  6
= lim
= -6
h0
h
h
10. 0 11. 2
12. DNE
t3
2
1
1t
t 1
= lim
= lim
= -.5
t  1 (t  1)(t  1)
t1 t  1
t 1
x6

15. sin( ) = 1
x6
2
16. DNE
17.
2  3a
18. lim (h + 3) = 3
h0
2  3a
19. DNE; in a day or two we'll further describe the way the limit doesn't
exist by saying that the limit is -.
20. ln(cos0) = ln1 = 0
21. DNE; in a day or two we'll further describe the way the limit doesn't
exist by saying that the limit is -.
4a
 2
22.
2a
23. DNE; in a day or two we'll further describe the way the limit doesn't
exist by saying that the limit is .
24. 1
solutions to part B.
1. cos(/2) = 0
2. sin(/2) = 1
3. The limit does not exist since the right and left sided limits do not agree.