If \(f(x) = \begin{cases} ax - 1 & \text{if } x \ge 1 \\ 2x + 1 & \text{if } x \lt 1 \end{cases}\) is continuous at \(x = 1\), then \(a\) equals
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4
For continuity at x = 1, the left-hand limit must equal the value f(1).
Since \(x = 1\) falls in the branch \(x \ge 1\):
\(f(1) = a - 1\)
For \(x \lt 1\), \(f(x) = 2x + 1\):
\( \displaystyle \lim_{x \to 1^-} f(x) = 2(1) + 1 = 3 \)
(The right-hand limit is \(a - 1\), the same as f(1).) Equating:
\(a - 1 = 3,\quad a = 4\)