If the function \( f(x)=\begin{cases}3x-6 & ,x\le 5\\4k & ,x\gt 5\end{cases} \) is continuous at \( x=5 \), then the value of \( k \) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle\frac{9}{4} \)
f is continuous at x = 5 when the left-hand limit, the right-hand limit and f(5) are equal.
\( \displaystyle f(5)=\lim_{x\to 5^-}(3x-6) \)
\( \displaystyle =15-6=9 \)
\( \displaystyle \lim_{x\to 5^+}f(x)=4k \)
\( \displaystyle 4k=9\ \Rightarrow\ k=\frac{9}{4} \)