Evaluate: `underset(xrarr0)lim((1-x)^n-1)/(x)`

Answer

To evaluate the limit

\[ \lim_{x \to 0} \frac{(1 - x)^n - 1}{x} \]

we can follow these steps:

### Step 1: Expand \( (1 - x)^n \) using the Binomial Theorem

Using the Binomial Theorem, we can expand \( (1 - x)^n \):

\[ (1 - x)^n = 1 - nx + \frac{n(n-1)}{2}x^2 - \frac{n(n-1)(n-2)}{6}x^3 + \ldots \]

### Step 2: Substitute the expansion into the limit

Now, substituting this expansion into our limit expression, we have:

\[ \lim_{x \to 0} \frac{(1 - x)^n - 1}{x} = \lim_{x \to 0} \frac{(1 - nx + \frac{n(n-1)}{2}x^2 - \ldots) - 1}{x} \]

### Step 3: Simplify the expression

The \(1\)s in the numerator cancel out:

\[ = \lim_{x \to 0} \frac{-nx + \frac{n(n-1)}{2}x^2 - \ldots}{x} \]

### Step 4: Factor out \(x\) from the numerator

Now we can factor \(x\) out of the numerator:

\[ = \lim_{x \to 0} \frac{x(-n + \frac{n(n-1)}{2}x - \ldots)}{x} \]

### Step 5: Cancel \(x\) from numerator and denominator

After canceling \(x\), we get:

\[ = \lim_{x \to 0} (-n + \frac{n(n-1)}{2}x - \ldots) \]

### Step 6: Evaluate the limit

As \(x\) approaches \(0\), all terms involving \(x\) vanish:

\[ = -n \]

Thus, the final answer is:

\[ \boxed{-n} \]

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