`lim_(xto0)((x-2)/(x+2))^(x+2)` equal to ?

Options

A. 0
B. 1
C. `e^(-4)`
D. `e-^(4)`

Answer

To solve the limit \( \lim_{x \to 0} \left( \frac{x-2}{x+2} \right)^{x+2} \), we will follow these steps:

### Step 1: Substitute \( x = 0 \) into the expression
First, we substitute \( x = 0 \) into the expression to evaluate the limit directly.

\[ \frac{x-2}{x+2} = \frac{0-2}{0+2} = \frac{-2}{2} = -1 \]

Now, substituting \( x = 0 \) into the exponent \( x + 2 \):

\[ x + 2 = 0 + 2 = 2 \]

Thus, we have:

\[ \left( \frac{x-2}{x+2} \right)^{x+2} = (-1)^{2} \]

### Step 2: Calculate the result
Now, we calculate:

\[ (-1)^{2} = 1 \]

### Conclusion
Therefore, the limit is:

\[ \lim_{x \to 0} \left( \frac{x-2}{x+2} \right)^{x+2} = 1 \]

### Final Answer
The answer is \( 1 \).

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