76 Higher Engineering Mathematics
Hence lim
x→0
x − sin x
x − tan x
= −
1
2
Now try the following exercise
Exercise 34 Further problems on limiting
values
Determine the following limiting values
1. lim
x→1
x 3 − 2x + 1
2x 3 + 3x − 5
1
9
2. lim
x→0
sin x
x
[1]
3. lim
x→0
ln(1 + x)
x
[1]
4. lim
x→0
x
2
− sin 3x
3x + x 2
[−1]
5. lim
θ→0
sin θ − θ cos θ
θ 3
1
3
6. lim
t →1
ln t
t 2 − 1
1
2
7. lim
x→0
sinh x − sin x
x 3
1
3
8. lim
θ→
π
2
sin θ − 1
ln sin θ
[1]
9. lim
t →0
sec t − 1
t sin t
1
2