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 x2
x x 1
5. Given f ( x)   , find
2 x1
a) lim f ( x)

b) lim f ( x)

c) lim f ( x)
x 1 x 1 x1
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?
x2
(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?
x4
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
x3
2x  4
r) lim
d) lim( x  x  1)
4
x  3  2 x
x 1
e) lim( y  1)(x  3) x2
s) lim
y 2 x  x3
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
x0 x2 w) lim
x  x 3  4 x  3
x2
j) lim
x 2 x  3 x  4
3 x2 1
x) lim
x  (3 x  2) 2
x5
k) lim
x 5 x 2  25
x2  x  2
l) lim
x 2 x2
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 x5
c) lim x  1
x 1
x3 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