Higher Engineering Mathematics

74 NUMBER AND ALGEBRA

Hence lim
x→ 0

{

x−sinx

x−tanx

}

=−

1

2

Now try the following exercise.

Exercise 38 Further problems on limiting

values

Determine the following limiting values

1. lim
   x→ 1

{

x^3 − 2 x+ 1

2 x^3 + 3 x− 5

}[

1

9

]

2. lim
   x→ 0

{

sinx

x

}

[1]

3. lim
   x→ 0

{

ln(1+x)

x

}

[1]

4. lim
   x→ 0

{

x^2 −sin 3x

3 x+x^2

}

[−1]

5. lim
   θ→ 0

{

sinθ−θcosθ

θ^3

}[

1

3

]

6. lim
   t→ 1

{

lnt

t^2 − 1

}[

1

2

]

7. lim
   x→ 0

{

sinhx−sinx

x^3

}[

1

3

]

8. lim
   θ→π 2

{

sinθ− 1

ln sinθ

}

[ 1 ]

9. lim
   t→ 0

{

sect− 1

tsint

}[

1

2

]