What is the value of `lim_(xtooo) ((x+6)/(x+1))^(x+4)`

Options

A. e
B. `e^(2)`
C. `e^(4)`
D. `e^(5)`

Explanation

To find the value of the limit

\[ \lim_{x \to \infty} \left(\frac{x + 6}{x + 1}\right)^{x + 4}, \]

we will break it down step by step.

### Step 1: Simplify the expression inside the limit
First, we can rewrite the fraction inside the limit:

\[ \frac{x + 6}{x + 1} = \frac{x + 1 + 5}{x + 1} = 1 + \frac{5}{x + 1}. \]

### Step 2: Rewrite the limit
Now, we can express the limit as:

\[ \lim_{x \to \infty} \left(1 + \frac{5}{x + 1}\right)^{x + 4}. \]

### Step 3: Use the exponential limit property
We know that

\[ \lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^{n} = e^a. \]

To apply this, we need to manipulate our expression to fit this form. We can rewrite the exponent \(x + 4\) in terms of \(x + 1\):

\[ x + 4 = (x + 1) + 3. \]

Thus, we can express our limit as:

\[ \lim_{x \to \infty} \left(1 + \frac{5}{x + 1}\right)^{(x + 1) + 3}. \]

### Step 4: Split the limit
This can be split into two parts:

\[ \lim_{x \to \infty} \left(1 + \frac{5}{x + 1}\right)^{x + 1} \cdot \left(1 + \frac{5}{x + 1}\right)^{3}. \]

### Step 5: Evaluate the first part
Using the exponential limit property:

\[ \lim_{x \to \infty} \left(1 + \frac{5}{x + 1}\right)^{x + 1} = e^5. \]

### Step 6: Evaluate the second part
Now, we evaluate the second part:

\[ \lim_{x \to \infty} \left(1 + \frac{5}{x + 1}\right)^{3}. \]

As \(x \to \infty\), \(\frac{5}{x + 1} \to 0\), so:

\[ \left(1 + \frac{5}{x + 1}\right)^{3} \to 1^{3} = 1. \]

### Step 7: Combine the results
Now, we combine both parts:

\[ e^5 \cdot 1 = e^5. \]

### Final Answer
Thus, the value of the limit is:

\[ \lim_{x \to \infty} \left(\frac{x + 6}{x + 1}\right)^{x + 4} = e^5. \]

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