Question

The value of $\displaystyle \lim_{x \to 0}\frac{(1-x)^n - 1}{x}$ is

• A. 0
• B. 1
• C. $-n$
• D. $n$

Hint:

For limits involving $(1-x)^n$, the first two terms of the binomial expansion are usually enough.

Answer 1

The Correct Option is C

Use the binomial expansion for small $x$:
\[ (1-x)^n = 1 - nx + \frac{n(n-1)}{2}x^2 + \cdots \] Then,
\[ (1-x)^n - 1 = -nx + O(x^2) \] So,
\[ \lim_{x\to 0} \frac{(1-x)^n - 1}{x} = \lim_{x\to 0} \frac{-nx + O(x^2)}{x} = -n \] Thus the limit equals $-n$.
Final Answer: $-n$