The value of `lim_(xtooo) ((100)/(1-x^(100))-(50)/(1-x^(50)))` is ___________.

Explanation

To solve the limit \( \lim_{x \to \infty} \left( \frac{100}{1 - x^{100}} - \frac{50}{1 - x^{50}} \right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the limit:
\[ \lim_{x \to \infty} \left( \frac{100}{1 - x^{100}} - \frac{50}{1 - x^{50}} \right) \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \( (1 - x^{100})(1 - x^{50}) \). We can rewrite the limit as:
\[ \lim_{x \to \infty} \left( \frac{100(1 - x^{50}) - 50(1 - x^{100})}{(1 - x^{100})(1 - x^{50})} \right) \]

### Step 3: Expand the numerator
Now, let's expand the numerator:
\[ 100(1 - x^{50}) - 50(1 - x^{100}) = 100 - 100x^{50} - 50 + 50x^{100} \]
This simplifies to:
\[ 50 + 50x^{100} - 100x^{50} \]

### Step 4: Substitute back into the limit
Now substituting back, we have:
\[ \lim_{x \to \infty} \frac{50 + 50x^{100} - 100x^{50}}{(1 - x^{100})(1 - x^{50})} \]

### Step 5: Analyze the limit as \( x \to \infty \)
As \( x \to \infty \), the terms \( x^{100} \) and \( x^{50} \) dominate the numerator and denominator. Thus, we can focus on the leading terms:
\[ \lim_{x \to \infty} \frac{50x^{100}}{-x^{150}} = \lim_{x \to \infty} \frac{50}{-x^{50}} = 0 \]

### Step 6: Conclusion
Thus, the value of the limit is:
\[ \boxed{0} \]