Question

If $f\left(x\right)=\left\{\begin{array}{cc}x+k,& x<3\\ 4,& x=3\\ 3x-5,& x>3\end{array}\right.$ is continuous at x = 3, then k = _____________.

Answer 1

The function $f\left(x\right)=\left\{\begin{array}{cc}x+k,& x<3\\ 4,& x=3\\ 3x-5,& x>3\end{array}\right.$ is continuous at x = 3.

$\therefore f\left(3\right)=\underset{x\to 3}{\lim }f\left(x\right)$

$\Rightarrow f\left(3\right)=\underset{x\to 3^{-}}{\lim }f\left(x\right)=\underset{x\to 3^{+}}{\lim }f\left(x\right)$ .....(1)

Now,

f(3) = 4 .....(2)

$\underset{x\to 3^{-}}{\lim }f\left(x\right)=\underset{x\to 3}{\lim }\left(x+k\right)=3+k$ .....(3)

$\underset{x\to 3^{+}}{\lim }f\left(x\right)=\underset{x\to 3}{\lim }\left(3x-5\right)=3\times 3-5=9-5=4$ .....(4)

From (1), (2), (3) and (4), we have

4 = 3 + k = 4

⇒ 3 + k = 4

⇒ k = 4 − 3 = 1

Thus, the value of k is 1.


If $f\left(x\right)=\left\{\begin{array}{cc}x+k,& x<3\\ 4,& x=3\\ 3x-5,& x>3\end{array}\right.$ is continuous at x = 3, then k = ___1___.