The value of `lim_(x to 1) ((p)/(1-x^(p))-(q)/(1-xq)),p,q,inN,` equals

Options

A. `-(3)/(4)`
B. 0 if n is even
C. `-(3)/(4)` if n is odd
D. none of these

Explanation

To find the limit

\[ \lim_{x \to 1} \left( \frac{p}{1 - x^p} - \frac{q}{1 - x^q} \right) \]

we can follow these steps:

### Step 1: Combine the fractions
We start by finding a common denominator for the two fractions:

\[ \frac{p}{1 - x^p} - \frac{q}{1 - x^q} = \frac{p(1 - x^q) - q(1 - x^p)}{(1 - x^p)(1 - x^q)} \]

### Step 2: Simplify the numerator
Now, we simplify the numerator:

\[ p(1 - x^q) - q(1 - x^p) = p - px^q - q + qx^p = (p - q) + qx^p - px^q \]

So we have:

\[ \frac{(p - q) + qx^p - px^q}{(1 - x^p)(1 - x^q)} \]

### Step 3: Evaluate the limit
Now we take the limit as \( x \to 1 \):

Substituting \( x = 1 \):

\[ \text{Numerator: } (p - q) + q(1) - p(1) = (p - q) + q - p = (p - q) + (q - p) = 0 \]

\[ \text{Denominator: } (1 - 1^p)(1 - 1^q) = 0 \cdot 0 = 0 \]

This gives us an indeterminate form \( \frac{0}{0} \), so we can apply L'Hôpital's Rule.

### Step 4: Apply L'Hôpital's Rule
Differentiate the numerator and denominator separately:

**Numerator:**
\[ \frac{d}{dx} \left( (p - q) + qx^p - px^q \right) = qpx^{p-1} - pqx^{q-1} \]

**Denominator:**
\[ \frac{d}{dx} \left( (1 - x^p)(1 - x^q) \right) = (1 - x^q)(-px^{p-1}) + (1 - x^p)(-qx^{q-1}) \]

### Step 5: Evaluate the limit again
Now we evaluate the limit again as \( x \to 1 \):

**Numerator:**
\[ q \cdot p \cdot 1^{p-1} - p \cdot q \cdot 1^{q-1} = qp - pq = 0 \]

**Denominator:**
\[ (1 - 1^q)(-p \cdot 1^{p-1}) + (1 - 1^p)(-q \cdot 1^{q-1}) = 0 \]

This is still \( \frac{0}{0} \), so we apply L'Hôpital's Rule again.

### Step 6: Differentiate again and evaluate
After differentiating again and simplifying, we find:

\[ \lim_{x \to 1} \frac{pq(p-1)x^{p-2} - pq(q-1)x^{q-2}}{pq(1^{q-1}) + p(q-1)x^{p-1} + q(p-1)x^{q-1}} \]

Evaluating this limit as \( x \to 1 \) gives us:

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

### Final Answer
Thus, the limit evaluates to:

\[ \frac{p - q}{2} \]