Sine Limit Examples
We begin by introducing the following two rules:
sin kx
kx
(i) lim
=1
and
(ii) lim
=1
x→0 kx
x→0 sin kx
For example we can say:
sin 5x
lim
=1
x→0 5x
or
Example 1
4 sin 4x
sin 4x
= lim
lim
x→0
x→0
x
4x
sin 4x
= 4 lim
x→0 4x
= 4(1)
= 4.
3θ
lim
=1
θ→0 sin 3θ
for k ∈ R and k 6= 0.
or
sin − 21 x
lim
=1
x→0
− 12 x
Multiplying top and bottom by 4
Applying rule (i) above
Example 2
6x
3x
= lim
Multiplying top and bottom by 2
lim
x→0 2 sin 6x
x→0 sin 6x
1
6x
=
lim
x→0
2
sin 6x
1
(1)
Applying rule (ii) above
=
2
1
=
.
2
Example 3
3x
3x
= lim sin 4x
lim
x→0
x→0 tan 4x
cos 4x
3x cos 4x
= lim
x→0 sin 4x
x = 3 lim (cos 4x)
x→0
sin 4x x = 3 lim cos 4x lim
x→0
x→0 sin 4x
1
4x
= 3 lim cos 4x
lim
x→0
4 x→0 sin 4x
3
=
.
4
Material developed by the Department of Mathematics & Statistics, N.U.I. Maynooth
and supported by the NDLR (www.ndlr.com).
1
Try the following exercises for practice:
(a)
sin 7x
x→0
x
lim
(b)
x
x→0 sin 8x
lim
(c)
lim
x→0
sin 7x
3x
(d)
tan t
t→0
t
lim
(e)
5 sin θ
θ→0
θ
lim
Solutions
(a) 7
(b) 81
(c) 73
(d) 1
(e) 5
2