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 $\underset{\mathit{x}\to 0^{-}}{\lim }$ f (x) = $\underset{\mathit{x}\to 0^{+}}{\lim }$ f (x) = f (0)
$\underset{\mathit{x}\to 0^{-}}{\lim }$ f (x) = $\underset{\mathit{x}\to 0}{\lim }\left[\mathit{s}\mathit{a}\mathit{i}\mathit{n}\frac{\mathit{\pi }}{2}\left(\mathit{x}+1\right)\right]$ = a sin $\frac{\mathit{\pi }}{2}$ = a (1) = a
$\underset{\mathit{x}\to 0^{+}}{\lim }$ f (x) = $\underset{\mathit{x}\to 0}{\lim }$ $\frac{\mathit{t}\mathit{a}\mathit{n}\mathit{x}-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{{\mathit{x}}^3}$ = $\underset{\mathit{x}\to 0}{\lim }$ $\frac{\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}-\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{{\mathit{x}}^3}$
= $\underset{\mathit{x}\to 0}{\lim }$ $\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}\left(1-\mathit{c}\mathit{o}\mathit{s}\mathit{x}\right)}{{\mathit{x}}^3\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$ = $\underset{\mathit{x}\to 0}{\lim }$ $\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}.2\mathit{s}\mathit{i}{\mathit{n}}^2\frac{\mathit{x}}{2}}{{\mathit{x}}^3\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$
= 2 $\underset{\mathit{x}\to 0}{\lim }$ $\frac{1}{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$ × $\underset{\mathit{x}\to 0}{\lim }$ $\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{\mathit{x}}$ × $\underset{\mathit{x}\to 0}{\lim }$ ${\left[\frac{\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{x}}{2}}{\mathit{x}}\right]}_0^2$
= 2 × 1 × 1 × $\frac{1}{4}$ × $\underset{\frac{\mathit{x}}{2}\to 0}{\lim }$ ${\left[\frac{\mathit{s}\mathit{i}\mathit{n}\frac{\mathit{x}}{2}}{\frac{\mathit{x}}{2}}\right]}^2$
= 2 × 1 × 1 × $\frac{1}{4}$ × 1 = $\frac{1}{2}$
Now, f(0) = a sin $\frac{\mathit{\pi }}{2}$ (0 + 1) = a sin $\frac{\mathit{\pi }}{2}$ = a × 1 = a
Since f is continuous at x = 0, a = $\frac{1}{2}$