# Solve this

Question:

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____________

Solution:

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___.