Question

If $g\left(x\right)$ is a polynomial satisfying $g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$ for all real $x$ and $y$ and $g\left(2\right)=5$, then $\underset{x\to 3}{\lim }g\left(x\right)=$

• A. $9$
• B. $10$
• C. $25$
• D. $20$

Answer 1

The correct option is B

$10$

Explanation for the correct option:

Step 1. Find the value of $g\left(1\right)$.

In the given equation $g\left(x\right)g\left(y\right)=g\left(x\right)+g\left(y\right)+g\left(xy\right)-2$, substitute $x=1$ and $y=2$.

$$\begin{gathered} g\left(1\right)g\left(2\right)=g\left(1\right)+g\left(2\right)+g\left(1·2\right)-2 \\ \Rightarrow g\left(1\right)\times 5=g\left(1\right)+5+5-2 \\ \Rightarrow 4g\left(1\right)=8 \\ \Rightarrow g\left(1\right)=2 \end{gathered}$$

Step 2. Find the value of $\underset{x\to 3}{\lim }g\left(x\right)$.

Now as $g\left(1\right)=2$ and $g\left(2\right)=5$ so one such function is $g\left(x\right)=x^2+1$.

Now for $x=3$

$\begin{array}{rcl}g\left(3\right)& =& 3^2+1\\ & =& 9+1\\ & =& 10\end{array}$

Thus the value of $\underset{x\to 3}{\lim }g\left(x\right)$ is

$\begin{array}{rcl}\underset{x\to 3}{\lim }g\left(x\right)& =& g\left(3\right)\\ & =& 10\end{array}$

Hence, the correct option is B.