8/2/2019 Math 36 B Unit 3 Supplementary Exercises

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MATH 36 BU nit 3 : Supplementary Exercises

I. A. Evaluate the following limits( if they exist).

8/2/2019 Math 36 B Unit 3 Supplementary Exercises

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II. Answer the following completely and neatly.

1. ( ) x lim x x  

→π

− +3

10 7 2

2.2

22

3 7 2lim

3 2 x

 x x

 x x→−

+ +

+ +

9. x 

lim x →−∞

 −    7

310

10.4 2

0

3 1lim

2 s

 s

 s s+

→

+

+

3.( )

h

 x h x lim

h→

+ −2 2

0

4.3

38

2 1lim

 x

 x

 x→−

−

5.1

1lim

1 x

 x

 x+→

−−

11.

( ) x 

 x lim

 x  x +

→

+    −  −

23

5 4

43

12.0

tanlimθ 

θ 

θ →

13.6

lim sec tan z 

 z z π →

 

6.   x 

 x lim

 x x →

−− −

2

24

16

3 10 814.

( )

t 

cos t  lim

t →

− 2

20

1 2

4

7.   x 

 x lim

 x −→ −

2

24 16

8.3

3

2lim

6 2 1 x

 x

 x x→+∞

−

− +

B. B. Consider the graph of   f   at

the right.

1. ( )lim x

 f x→ +∞

=  

2. ( )1

lim x

 f x−

→

=  

3. ( )1

lim x

 f x+

→−

=

4. ( )1

lim x

 f x−

→−

=

5. ( )1

lim x

 f x→−

=

6. ( )2

lim x

 f x→

=

7. f  has removable

discontinuity atA

8. f  has essential discontinuity

at

A

15.2 2

lim2 1 x

 x x

 x→−∞

+

−

16.( )

2

0

6 4lim

sin 2 y

 y y

 y→

−

8/2/2019 Math 36 B Unit 3 Supplementary Exercises

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1. Prove using definitionδ − ε that ( ) x alim mx b ma b, m

→

+ = + ≠ 0.

2. Given ( ) ( ) 223 −=+= x x g and  x x f    

Is  g  continuous at 2? Why?

Is  f g o continuous at 2? Why?

Is  g f  o continuous at -10? Why?

3. If  ( ) 1+= x x f  

(a) Is  f  continuous on [ )+∞− ,1 ?

(b) Is  f  continuous [ )+∞−∈∀ ,1 x ?

4. Prove that the following functions are discontinuous at thegiven number a. Then, determine the type of discontinuity. If applicable,

redefine ( )f a to remove the discontinuity. ( 3 points each )

i. ( )x if x  

f x a x if x 

− <= = + ≥

24 112 3 1

ii. ( )x 

g x a x 

−= =

−11 22

22

5. Given ( )

>

≤<<−

=

9

925

213

 x x

 x

 x x

 x g  . Determine if  g is continuous on

the following intervals

( )2,

∞− ( ]2,

∞− ( )+∞,3

[ )+∞,3

( )+∞∞−,

6. Determine for each of the indicated intervals whether thefunction is continuous on that interval. If your answer is no, explain why. ( 9points )

( ) x x 

 x f x  x 

 x 

−≠ ±= −

=

3

2

42

4

2 2

( ],0 4   ( ),−2 2   ( ],−∞ −2   ( ),+∞2   ( ],−2 2   ( ),−∞ +∞

7. Find the values of s and d that will make the function

continuous on ( ),−∞ +∞ . ( )

 x x 

g x sx d x  

 x x 

≤= + < <− ≥

1

1 4

2 4

8. State the Intermediate Value Theorem.