Find the value of a for which the function `f` defined by `f(x)={asinpi/2(x+1), ,xlt=0(tanx-sinx)/(x^3),x >0"` is continuous at x=0

Explanation

To find the value of \( a \) for which the function \( f \) is continuous at \( x = 0 \), we need to ensure that the left-hand limit, right-hand limit, and the function value at \( x = 0 \) are all equal.

The function \( f \) is defined as:
\[
f(x) =
\begin{cases}
a \sin\left(\frac{\pi}{2}(x + 1)\right) & \text{if } x \leq 0 \\
\frac{\tan x - \sin x}{x^3} & \text{if } x > 0
...