To solve this problem, we need to manipulate the given equation to find the value of the expression:

Given:

\(\frac{1}{x} = \frac{1}{p} + \frac{1}{q}\)

First, find a common denominator on the right-hand side:

\(\frac{1}{x} = \frac{q + p}{pq}\)

This implies:

\(x = \frac{pq}{p + q}\)

Substitute \(x\) in the given expression:

\(\frac{pq}{p^2-q^2}\left(\frac{x+p}{x-p}-\frac{x+q}{x-q}\right)\)

This equals:

\(\frac{pq}{p^2-q^2}\left(\frac{\left(\frac{pq}{p+q}\right)+p}{\left(\frac{pq}{p+q}\right)-p}-\frac{\left(\frac{pq}{p+q}\right)+q}{\left(\frac{pq}{p+q}\right)-q}\right)\)

First handle the two fractions separately:

• Let's simplify the first part: 
  \(\frac{\left(\frac{pq}{p+q}\right)+p}{\left(\frac{pq}{p+q}\right)-p} = \frac{\frac{pq + p(p + q)}{p+q}}{\frac{pq - p(p+q)}{p+q}}\)
• Now simplify the second part: 
  \(\frac{\left(\frac{pq}{p+q}\right)+q}{\left(\frac{pq}{p+q}\right)-q} = \frac{\frac{pq + q(p + q)}{p+q}}{\frac{pq - q(p+q)}{p+q}}\)

Subtract these simplified terms:

\(-\frac{2pq + p^2}{p^2} + \frac{2pq + q^2}{q^2} = -1 +1\)

Finally, plug these back into the main expression:

\(\frac{pq}{p^2-q^2}\times 2 = 2\)

The correct answer is:

Option 2