`lim_(x to infty) [1+ 2//x)]^(x)` equals:

Options

A. e
B. `infty`
C. `e^(2)`
D. `1//e`

Answer

To solve the limit \( \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x \), we can use a well-known limit result. Here’s the step-by-step solution:

### Step 1: Identify the limit form
We start with the limit:
\[ \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x \]
As \( x \) approaches infinity, \( \frac{2}{x} \) approaches 0.

### Step 2: Recognize the standard limit
We can relate this limit to a standard limit form:
\[ \lim_{x \to \infty} \left(1 + \frac{\lambda}{f(x)}\right)^{f(x)} = e^\lambda \]
where \( f(x) \to \infty \) as \( x \to \infty \) and \( \lambda \) is a constant.

### Step 3: Assign values to \( \lambda \) and \( f(x) \)
In our case:
- \( \lambda = 2 \)
- \( f(x) = x \)

As \( x \to \infty \), \( f(x) \) indeed approaches infinity.

### Step 4: Apply the limit result
Using the limit result:
\[ \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x = e^2 \]

### Step 5: Conclusion
Thus, we find that:
\[ \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^x = e^2 \]

### Final Answer
The final answer is:
\[ e^2 \]

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