Question

Let a function $f\left(x\right)=\{\begin{array}{c}bx+cforx>1\\ 3cx-2b+1forx<1\end{array}$. Then a relation between $b$ and $c$ such that $\underset{x\to 1}{lim}f\left(x\right)$ exists is

• A. $3b+2c+1=0$
• B. $3b+2c-1=0$
• C. $3b-2c-1=0$
• D. $b-2c+1=0$

Answer 1

The correct option is C $3b-2c-1=0$

$\underset{x\to 1}{lim}f\left(x\right)$ exists

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

$\Rightarrow \underset{x\to 1+}{lim}bx+c=\underset{x\to 1-}{lim}(3cx-2b+1)$

$\Rightarrow b+c=3c-2b+1$

$\Rightarrow 3b-2c-1=0$