2.1 Limits of Functions by SGLee - HSKim- SWSun
1-4. Find the following limits or explain why the limit does not exist.
1. 
Calculus-Sec-2-1-Solution
2.1 Limits of Functions by SGLee - HSKim- SWSun
1-4. Find the following limits or explain why the limit does not exist.
1.
2. 
If 
exists, then 
, when 
approaches to 0
(1)
(2)
, so 
than 
does not exist when 
approaches to 0. 
does not exist.
3.
http://matrix.skku.ac.kr/cal-lab/cal-2-1-3.html
We draw 
with the following Sage command. We see it diverge (
) as 
.
We can find 
with the following Sage command.
+Infinity
4. 
Since 
when 
, 
7. 
http://matrix.skku.ac.kr/cal-lab/cal-2-1-7.html
We draw 
with the following Sage command. We see the function converge to 
as 
.
We can find 
with the following Sage command.
8. The sign function, denoted by 
, is defined by the following piecewise function.
Find the following limits or explain why the limit does not exist.
(a) 
(b) 
does not exist (
)
(c) 
(d) 
9. Consider the function 
.
(a) Find 
and 
.
(b) Find the asymptotes of 
; vertical, horizontal, vertical and oblique. (For the oblique asymptote, find the straight line 
which is closer and closer to 
as 
)
: vertical asymptote
: oblique asymptote
(c) Sketch the graph.
10. Draw the graph of a function 
with all of the following properties:
(a) its domain is 
(b) there is a vertical asymptote at 
(c) 
(d) 
(e) 
does not exist.
(f) 
does not exist.
(g) 
11. Let 
.
(a) Find 
or explain why the limit does not exist.
.
(b) Find 
and 
such that 
for all 
.
Since 
, we have 
.
Take, 
(c) Use the Squeeze Theorem to find 
.
We know that 
and 
, so by the Squeeze Theorem,
13. Use squeeze Theorem to find 
.
http://matrix.skku.ac.kr/cal-lab/cal-2-1-13.html
(※
is not always smaller than 
, but it is when near 0.
So the Squeeze Theorem can be used.)
14. Let 
. Find all positive integer 
such that 
ⅰ) 
; 
ⅱ) 
, 
(
is positive integer) ; 
ⅲ) 
, 
(
is positive integer) ; 
ⅳ) 
; 
Therefore 
When 
.
15. Find all the asymptotes (vertical, horizontal, and oblique) of the function 
.
http://matrix.skku.ac.kr/cal-lab/cal-2-1-15.html
, so 
,
,
and
Thus, x=-1 and x=2 are vertical asymptotes.
, Thus, 
is an oblique asymptote.
16. Find the limit 
.
17. Consider 
.
(a) Find all the vertical asymptotes for 
.
, so
. 
Thus, 
are vertical asymptotes.
(b) If we restrict the domain to 
, then show that there exists an inverse function defined on 
.
(c) If the above inverse function is 
, then find all the horizontal asymptotes.
18. Find 
such that 
whenever 
.
19. Use an 
argument to prove that 
.
Let 
be a given positive number. Here 
and 
. Claim is to find a number 
such that 
whenever 
.
With easy computation, we may choose 
to get the desired result.
20. Use the 
argument to prove that 
if 
.
Let 
be a given positive number. Here 
and 
. Claim is to find a number 
such that 
whenever 
.
With an easy computation, we may choose 
to get the desired result.
21. 
[Find 
] Let 
. If 
, then 
=
= 
= 
.
[Side calculation] 
. ■
22. 
Let 
be a given positive number. Here 
and 
. Claim is to find a number 
such that 
whenever 
.
With an easy computation, we may choose 
to get the desired result.
23. 
[Find 
] Let 
= 
. If |
| < 
, then 
= 
.
[Side calculation] 
. ■
24. 
[Find 
] Let 
= 
. If 
, then 
=
=
=
= 
.
[Side calculation] 
. 
. ■
25. 
[Find 
] Let 
{
}. If 
, then 
=
[Side calculation] 
. 
. ■
26. 
[Find 
] Let 
. If 
, then 
=
.
[Side calculation] 
Take 
. If 
, then 
. 
=
. ■
27. 
.
http://matrix.skku.ac.kr/cal-lab/cal-2-1-27.html
[Find 
] Let 
. If 
, then
=
.
[Side calculation] 
and 
. 
. ■
28. 
.
http://matrix.skku.ac.kr/cal-lab/cal-2-1-28.html
, [Find 
] Let 
(
). If 
, then 
.
[Side calculation] Since 
, 
. 
. ■
29. 
Given any (large) number 
we must find 
such that 
whenever 
. Since both 
and 
are positive, 
whenever 
.
Taking the square root of both sides and recalling that 
, we get 
whenever 
.
So for any 
, choose 
.
Now if 
, then 
, that is,
.
Thus 
whenever 
.
Therefore 
.
30. 
Let 
be a given positive number. Here 
and 
. Claim is to find a number 
such that 
whenever 
.
With an easy computation, we may choose 
to get the desired result.
31. Use an 
argument to prove that 
Given any (large) number 
to find 
such that 
whenever 
.
Since both 
and 
are positive, 
whenever 
.
So for any 
, choose 
.
Now if 
, then 
Thus 
32. If 
and 
, where 
is a real number. Show that
(a) 
(b) 
if 
(a) 
(b) 
6. 
12. Use the Squeeze Theorem to find 
.
