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.

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.

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,

12. Use the Squeeze Theorem to find .

13. Use squeeze Theorem to find .

(※ 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 .

, 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)