2.1 Limits of functions

1-4. Find the following limits or explain why the limit does not exist.

1.

2.

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.

5.

http://matrix.skku.ac.kr/cal-lab/cal-2-1-5.html

6.

http://matrix.skku.ac.kr/cal-lab/cal-2-1-6.html

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.

0

8. The sign function, denoted by , is defined by the following formulas

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 .

so,

(c) Use the Squeeze Theorem to find .

We know that and , so by the Squeeze Theorem,

12. Use the Squeeze Theorem to find .

http://matrix.skku.ac.kr/cal-lab/cal-2-1-12.html

13. Use squeeze Theorem to find .

http://matrix.skku.ac.kr/cal-lab/cal-2-1-13.html

(¡Ø 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

¥¡) n=4 ;

¥¢) n<4 ;

¥£) n>4 ;

Therefore

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.

21-26. Prove the statements using an argument.

21.

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.

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.

24.

25.

Given any (large) number to 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 .

26.

27. 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

28. If and , where is a real number. Show that

(a)

(b) if

(a)

(b)