Pair Of Linear Equations In Two Variables

To solve the problem, we need to analyze the functional equation and evaluate the given limit condition. Our aim is to find the value of .

First, consider the functional equation given:

Let's substitute with to get another equation:

Now, let's denote these equations as:

• Equation (1):
• Equation (2):

We solve these equations simultaneously. Add Equation (1) and Equation (2):

Simplify:

Rearrange to find :

Simplify further:

For the condition:

As , the component must also tend to zero for any finite value of . This indicates that the singularity should be removed by the function, suggesting .

Thus, we need:

under the condition:

yields

Solving, we find , leading to:

Therefore,

Finally,

Thus, the answer is correct:

4.