Suppose f(x) = and iff(x) = f(1) what are possible values of a and b?
Question:

Suppose $f(x)=\left\{\begin{array}{ll}a+b x, & x<1 \\ 4, & x=1 \\ b-a x & x>1\end{array}\right.$ and if $\lim _{x \rightarrow 1} f(x)=f(1)$ what are possible values of $a$ and $b ?$

Solution:

The given function is

$f(x)= \begin{cases}a+b x, & x<1 \\ 4, & x=1 \\ b-a x & x>1\end{cases}$

$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1}(a+b x)=a+b$

$\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1}(b-a x)=b-a$

$f(1)=4$

It is given that $\lim _{x \rightarrow 1} f(x)=f(1)$.

$\therefore \lim _{x \rightarrow 1^{-}} f(x)=\lim _{\left.x \rightarrow\right|^{+}} f(x)=\lim _{x \rightarrow 1} f(x)=f(1)$

$\Rightarrow a+b=4$ and $b-a=4$

On solving these two equations, we obtain $a=0$ and $b=4$.

Thus, the respective possible values of $a$ and $b$ are 0 and 4

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