Evaluate the following limits .
`Lim_(x to 0 ) ((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: Substitute \( t = 1 + x \)

We start by substituting \( t = 1 + x \). As \( x \to 0 \), \( t \to 1 \). Thus, we rewrite the limit in terms of \( t \):

\[ x = t - 1 \quad \text{and} \quad \lim_{x \to 0} \text{ becomes } \lim_{t \to 1} \]

Now, our limit becomes:

\[ \lim_{t \to 1} \frac{t^n - 1}{t - 1} \]

### Step 2: Apply the limit property

We can apply the limit property which states:

\[ \lim_{x \to a} \frac{x^n - a^n}{x - a} = n a^{n-1} \]

In our case, \( a = 1 \) and \( n \) is the exponent. Therefore, we can apply this property:

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

### Conclusion

Thus, we find that:

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

### Final Answer

The value of the limit is:

\[ \boxed{n} \]

---