Evaluate the following limits :
`lim_(x to 0)((x+1)^(5)-1)/x`

Answer

To evaluate the limit

\[ \lim_{x \to 0} \frac{(x+1)^5 - 1}{x}, \]

we will follow these steps:

### Step 1: Check the form of the limit
First, we substitute \(x = 0\) into the expression:

\[ \frac{(0+1)^5 - 1}{0} = \frac{1 - 1}{0} = \frac{0}{0}. \]

Since we have an indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule.

**Hint for Step 1:** Always check the limit by direct substitution first to identify if it is an indeterminate form.

### Step 2: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, if we have a limit of the form \( \frac{0}{0} \), we can differentiate the numerator and the denominator separately:

\[ \lim_{x \to 0} \frac{(x+1)^5 - 1}{x} = \lim_{x \to 0} \frac{\frac{d}{dx}((x+1)^5 - 1)}{\frac{d}{dx}(x)}. \]

### Step 3: Differentiate the numerator and denominator
Now we differentiate the numerator:

- The derivative of \((x+1)^5\) using the chain rule is:

\[ 5(x+1)^4 \cdot \frac{d}{dx}(x+1) = 5(x+1)^4 \cdot 1 = 5(x+1)^4. \]

- The derivative of \(1\) is \(0\).

So, the derivative of the numerator is:

\[ 5(x+1)^4. \]

The derivative of the denominator \(x\) is simply \(1\).

### Step 4: Rewrite the limit
Now we can rewrite the limit using these derivatives:

\[ \lim_{x \to 0} \frac{5(x+1)^4}{1} = \lim_{x \to 0} 5(x+1)^4. \]

### Step 5: Substitute \(x = 0\) again
Now we substitute \(x = 0\) into the limit:

\[ 5(0+1)^4 = 5 \cdot 1^4 = 5 \cdot 1 = 5. \]

### Final Result
Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{(x+1)^5 - 1}{x} = 5. \]