Evaluate `lim_(x to 1//4) (4x - 1)/(2sqrt(x) - 1)`

Explanation

To evaluate the limit

\[ \lim_{x \to \frac{1}{4}} \frac{4x - 1}{2\sqrt{x} - 1}, \]

we will follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( x = \frac{1}{4} \) into the expression to check if we get an indeterminate form.

\[ 4\left(\frac{1}{4}\right) - 1 = 1 - 1 = 0, \]
\[ 2\sqrt{\frac{1}{4}} - 1 = 2 \cdot \frac{1}{2} - 1 = 1 - 1 = 0. \]

Since both the numerator and denominator approach 0, we have an indeterminate form of type \( \frac{0}{0} \).

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form, we can apply L'Hôpital's Rule, which states that if we have \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), we can take the derivative of the numerator and the derivative of the denominator.

**Differentiate the numerator:**
\[ \frac{d}{dx}(4x - 1) = 4. \]

**Differentiate the denominator:**
\[ \frac{d}{dx}(2\sqrt{x} - 1) = 2 \cdot \frac{1}{2\sqrt{x}} = \frac{1}{\sqrt{x}}. \]

### Step 3: Rewrite the limit
Now we can rewrite the limit using the derivatives we found:

\[ \lim_{x \to \frac{1}{4}} \frac{4}{\frac{1}{\sqrt{x}}}. \]

### Step 4: Substitute the limit value again
Now substitute \( x = \frac{1}{4} \) into the new expression:

\[ \sqrt{\frac{1}{4}} = \frac{1}{2} \Rightarrow \frac{1}{\sqrt{\frac{1}{4}}} = 2. \]

Thus, we have:

\[ \lim_{x \to \frac{1}{4}} \frac{4}{\frac{1}{\sqrt{x}}} = 4 \cdot 2 = 8. \]

### Final Answer
Therefore, the limit is

\[ \boxed{8}. \]