If f(x)=≤ft […] . is continuous at x = 1, then a equals

The correct answer is Option (3) → 4

We are given:

$f(x)=\left\{\begin{matrix}ax-1& if\, x ≥1\\2x+1& if\,x < 1\end{matrix}\right.$

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