If $f(x)=\left\{\begin{array}{cl}a x+1, & \text { if } x \geq 1 \\ x+2, & \text { if } x<1\end{array}\right.$ is continuous, then 'a' should be equal to____________
It is given that, the function $f(x)=\left\{\begin{array}{cl}a x+1, & \text { if } x \geq 1 \\ x+2, & \text { if } x<1\end{array}\right.$ is continuous.
So, the function f(x) is continuous at x = 1.
$\therefore f(1)=\lim _{x \rightarrow 1} f(x)$
$\Rightarrow f(1)=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)$ .....(1)
Now,
$f(1)=a \times 1+1=a+1$ .....(2)
$\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1}(a x+1)=a \times 1+1=a+1$ .....(3)
$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1}(x+2)=1+2=3$ ....(4)
From (1), (2), (3) and (4), we have
$a+1=3=a+1$
$\Rightarrow a=3-1=2$
Thus, the value of a is 2.
If $f(x)=\left\{\begin{array}{cl}a x+1, & \text { if } x \geq 1 \\ x+2, & \text { if } x<1\end{array}\right.$ is continuous, then ' $a$ ' should be equal to ___2___.