If \(g(x) = \begin{cases}\displaystyle\frac{\alpha x}{|x|}, & \text{if } x \lt 0\\ 5, & \text{if } x \ge 0\end{cases}\) is continuous at \(x = 0\), then the value of \(\alpha\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → −5
For continuity at x = 0, the left-hand limit, the right-hand limit and g(0) must all be equal.
For x ≥ 0, g(x) = 5, so the right-hand limit and g(0) are both 5.
For x < 0, |x| = −x, so
\(\displaystyle g(x) = \frac{\alpha x}{-x} = -\alpha\)
\(\displaystyle \lim_{x \to 0^-} g(x) = -\alpha\)
Equating with 5:
\(-\alpha = 5,\quad \alpha = -5\)