What is `lim_(xto0)((1+x)^(n)-1)/(x)` equal to ?

Options

A. 0
B. 1
C. n
D. `n-1`

Answer

To solve the limit \( \lim_{x \to 0} \frac{(1+x)^n - 1}{x} \), we will follow these steps:

### Step 1: Substitute \( y = 1 + x \)
We start by substituting \( y = 1 + x \). As \( x \) approaches 0, \( y \) approaches 1. Thus, we can rewrite the limit in terms of \( y \):

\[ x = y - 1 \]

Now, we can rewrite the limit:

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

### Step 2: Recognize the limit form
The expression \( \frac{y^n - 1}{y - 1} \) is in the form of a limit that can be evaluated using the derivative definition or the formula for limits. This is a standard limit that can be evaluated as follows:

\[ \lim_{y \to 1} \frac{y^n - 1}{y - 1} = n \cdot 1^{n-1} = n \]

### Step 3: Conclusion
Thus, we find that:

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

### Final Answer:
The limit is equal to \( n \).

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