Evaluate : `lim_(x to 4 ) (sqrt(x)-2)/(x-4)`

Answer

To evaluate the limit \( \lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} \), we can follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( x = 4 \) into the expression:
\[ \frac{\sqrt{4} - 2}{4 - 4} = \frac{2 - 2}{0} = \frac{0}{0} \]
This results in an indeterminate form \( \frac{0}{0} \), so we need to manipulate the expression further.

**Hint:** When you encounter \( \frac{0}{0} \), it indicates that you need to simplify the expression.

### Step 2: Rewrite the denominator
We can rewrite the denominator \( x - 4 \) as \( \sqrt{x}^2 - 2^2 \). This allows us to use the difference of squares:
\[ x - 4 = \sqrt{x}^2 - 2^2 = (\sqrt{x} - 2)(\sqrt{x} + 2) \]

**Hint:** Use the identity \( a^2 - b^2 = (a - b)(a + b) \) to factor the denominator.

### Step 3: Substitute the factorization into the limit
Now we can rewrite the limit:
\[ \lim_{x \to 4} \frac{\sqrt{x} - 2}{(\sqrt{x} - 2)(\sqrt{x} + 2)} \]
We can cancel \( \sqrt{x} - 2 \) from the numerator and the denominator (as long as \( x \neq 4 \)):
\[ \lim_{x \to 4} \frac{1}{\sqrt{x} + 2} \]

**Hint:** Cancel common factors to simplify the limit expression.

### Step 4: Evaluate the limit
Now we can substitute \( x = 4 \) into the simplified expression:
\[ \frac{1}{\sqrt{4} + 2} = \frac{1}{2 + 2} = \frac{1}{4} \]

**Hint:** After simplification, substitute the limit value directly into the expression.

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} = \frac{1}{4} \]