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'll break it down step by step.

### Step 1: Analyze the individual terms

As \( x \to \infty \), both \( x^{100} \) and \( x^{50} \) approach infinity. Therefore, we can rewrite the terms:

\[ 1 - x^{100} \to -x^{100} \quad \text{and} \quad 1 - x^{50} \to -x^{50}. \]

### Step 2: Rewrite the limit

Substituting these approximations into the limit gives:

\[ \lim_{x \to \infty} \left( \frac{100}{-x^{100}} - \frac{50}{-x^{50}} \right) = \lim_{x \to \infty} \left( -\frac{100}{x^{100}} + \frac{50}{x^{50}} \right). \]

### Step 3: Simplify the expression

This can be simplified to:

\[ \lim_{x \to \infty} \left( -\frac{100}{x^{100}} + \frac{50}{x^{50}} \right). \]

### Step 4: Evaluate the limit

Now, we evaluate each term separately as \( x \to \infty \):

1. The term \( -\frac{100}{x^{100}} \) approaches \( 0 \) because the denominator grows much faster than the numerator.
2. The term \( \frac{50}{x^{50}} \) also approaches \( 0 \) for the same reason.

Thus, we have:

\[ \lim_{x \to \infty} \left( -\frac{100}{x^{100}} + \frac{50}{x^{50}} \right) = 0 + 0 = 0. \]

### Final Answer

Therefore, the value of the limit is:

\[ \boxed{0}. \]