If $f(x)=\left\{\begin{matrix}ax-1& if\, x ≥1\\2x+1& if\,x < 1\end{matrix}\right.$ is continuous at $x = 1$, then $a$ equals

4

The correct answer is Option (3) → 4

We are given:

f(x) = {

 ax − 1  if x ≥ 1

 2x + 1  if x < 1

}

To ensure continuity at x = 1, we require:

lim_x→1⁻ f(x) = lim_x→1⁺ f(x) = f(1)

Left-hand limit:

lim_x→1⁻ f(x) = lim_x→1⁻ (2x + 1) = 2(1) + 1 = 3

Right-hand limit:

lim_x→1⁺ f(x) = lim_x→1⁺ (ax − 1) = a(1) − 1 = a − 1

For continuity:

a − 1 = 3 ⇒ a = 4