PMB 3004: Calculus

Tutorial 3

1. Use your calculator to complete the table, and use your results to estimate the given limit.

a) 1

12lim

2

1

x

xx

x

x 0.9 0.99 0.999 1.001 1.01 1.1

)(xf

b) 1

14lim

2

1

x

xx

x

x 0.9 0.99 0.999 1.001 1.01 1.1

)(xf

4. Given

26

2)(

2

x x

xxxf , find )(lim

2xf

x

5. Given

12

1)(

x

x xxf , find

a) )(lim1

xfx

b) )(lim1

xfx

c) )(lim1

xfx

6. Let

.2,12

2,3)(

xxxx

xf

3

02 4

y

x

xy 3 12 xy

(a) Find )(lim2

xfx

and )(lim2

xfx

.

(b) Does )(lim2

xfx

exist? If so, what is it? If not, why not?

(c) Find )(lim4

xfx

and )(lim4

xfx

.

(d) Does )(lim4

xfx

exist? If so, what is it? If not, why not?

7. Find the limits for each of the following below:

a) 5lim3x

b) cx 1lim

c) 3

3lim xx

d) )1(lim 4

1

xx

x

e) )3)(1(lim2

xyy

f) )(lim 3

2yy

y

g) x

x

x

43lim

2

2

h) 4

32lim

3

2

2

x

xx

x

i) 20

1lim

xx

j) 43

2lim

32

xx

x

x

k) 25

5lim

25

x

x

x

l) 2

2lim

2

2

x

xx

x

m) 3

9lim

2

3

t

t

t

n) 224

1lim

2

2

1

xx

x

x

o) x

x

x

9)3(lim

2

0

p) h

h

h

4)2(lim

2

0

q) 1

2lim

xx

r) x

x

x 23

42lim

s) 3

2lim

x

x

x

t) 1

lim2

3

r

r

r

u) 74

125lim

2

t

tt

t

v) 185

243lim

3

2

xx

xx

x

w) 34

1lim

3

2

xx

x

x

x) 2

2

)23(

1lim

x

x

x

8. Find the limits.

a)34

3lim23

xx

xx

c) 23

1lim1

x

xx

b)5

103lim2

5

xxx

x.

d)9

3lim

9

x

xx

.

9. Show that the function

4

6)(2

2

x

xxxf continuous

at 3x .

10. Show that the function

.1 if3

,1 if2

,1 if2

)(

xx

xx

xx

xf

is continuous at 1x