f (x) =
The given function f is defined for all x ∊ R.
It is known that a function f is continuous at x = 0, if

lim

x→0−

f (x) =

lim

x→0+

f (x) = f (0)

lim

x→0−

f (x) =

lim

x→0

[sain

π

2

(x+1)] = a sin

π

2

= a (1) = a

lim

x→0+

f (x) =

lim

x→0

tanx−sinx

x3

=

lim

x→0

sinx cosx −sinx

x3

=

lim

x→0

sinx(1−cosx)

x3cosx

=

lim

x→0

sinx.2sin2 x 2

x3cosx

= 2

lim

x→0

1

cosx

×

lim

x→0

sinx

x

×

lim

x→0

[

sin x 2

x

]02
= 2 × 1 × 1 ×

1

4

×

lim

x 2 →0

[

sin x 2

x 2

]2
= 2 × 1 × 1 ×

1

4

× 1 =

1

2

Now, f(0) = a sin

π

2

(0 + 1) = a sin

π

2

= a × 1 = a
Since f is continuous at x = 0, a =

1

2