If the function \(f(x) = \begin{cases} ax + 3, & x \le 2\\ x^{2} + 5, & x \gt 2\end{cases}\) is continuous at \(x = 2\), then the value of a is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 3
For continuity at x = 2, the left-hand limit, the right-hand limit and f(2) must be equal.
\( \displaystyle \lim_{x \to 2^-} f(x) = f(2) = 2a + 3 \)
\( \displaystyle \lim_{x \to 2^+} f(x) = 2^{2} + 5 = 9 \)
Equating the two:
\(2a + 3 = 9\)
\(a = 3\)