`lim_(xrarr2)(sqrt(x+2)-sqrt(x-2))/(x+2)` is equal to

Options

A. 2
B. 1
C. `1//2`
D. 0

Answer

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

### Step 1: Substitute the limit value
We start by substituting \( x = 2 \) directly into the limit expression:

\[ \frac{\sqrt{2+2} - \sqrt{2-2}}{2+2} = \frac{\sqrt{4} - \sqrt{0}}{4} \]

### Step 2: Simplify the expression
Now, we simplify the expression:

\[ \frac{2 - 0}{4} = \frac{2}{4} \]

### Step 3: Final calculation
Now, we can simplify \( \frac{2}{4} \):

\[ \frac{2}{4} = \frac{1}{2} \]

### Conclusion
Thus, the limit is:

\[ \lim_{x \to 2} \frac{\sqrt{x+2} - \sqrt{x-2}}{x+2} = \frac{1}{2} \]