1. PMB 3004: Calculus
Tutorial 3
1. Use your calculator to complete the table, and use your results to estimate the given limit.
2x 2 x 1
a) lim
x 1 x 1
x 0.9 0.99 0.999 1.001 1.01 1.1
f (x)
4x 2 x 1
b) lim
x 1 x 1
x 0.9 0.99 0.999 1.001 1.01 1.1
f (x)
x2 x 2
4. Given f ( x) , find lim f ( x)
6 x x 2 x2
x x 1
5. Given f ( x) , find
2 x1
a) lim f ( x)
b) lim f ( x)
c) lim f ( x)
x 1 x 1 x1
3 x, x 2
6. Let f ( x) x
2 1, x 2.
y
3 y 3 x y x 1
2
0 x
2 4
2. (a) Find lim f ( x) and lim f ( x) .
x 2 x 2
(b) Does lim f ( x) exist? If so, what is it? If not, why not?
x2
(c) Find lim f ( x) and lim f ( x) .
x 4 x 4
(d) Does lim f ( x) exist? If so, what is it? If not, why not?
x4
7. Find the limits for each of the following below:
a) lim 5
x 3
( 2 h) 2 4
p) lim
h 0 h
b) lim c
x 1 2
q) lim
x x 1
c) lim x 3
x3
2x 4
r) lim
d) lim( x x 1)
4
x 3 2 x
x 1
e) lim( y 1)(x 3) x2
s) lim
y 2 x x3
f) lim( y 3 y) r3
y 2 t) lim
r r 2 1
3x 2 4
g) lim
x 2
5t 2 2t 1
x u) lim
t 4t 7
2x2 x 3
h) lim 3 4x 2x 2
x 2 x3 4 v) lim
x 5 x 3 8 x 1
1
i) lim x2 1
x0 x2 w) lim
x x 3 4 x 3
x2
j) lim
x 2 x 3 x 4
3 x2 1
x) lim
x (3 x 2) 2
x5
k) lim
x 5 x 2 25
x2 x 2
l) lim
x 2 x2
t2 9
m) lim
t 3 t 3
x2 1
n) lim
x 1 4 x 2 2 x 2
( x 3) 2 9
o) lim
x 0 x
3. 8. Find the limits.
a) lim 2 x 3 b) lim x 3x 10 .
2
x 3 x 4 x 3 x 5 x5
c) lim x 1
x 1
x3 2 d) lim x 3 .
x 9 x 9
9. Show that the function
f ( x) x x 6 continuous
2
x2 4
at x 3 .
10. Show that the function
x 2 if x 1,
f ( x ) 2 x if x 1,
x 3 if x 1.
is continuous at x 1