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You uploaded an image containing the following limit expression:

\[
\lim_{x \to 1} \frac{x^2 - 1}{\sqrt{x + 3} - \sqrt{4x}}
\]

Let's solve it step by step.

### Step 1: Factorize the numerator
The numerator \( x^2 - 1 \) is a difference of squares, so it can be factored as:

\[
x^2 - 1 = (x - 1)(x + 1)
\]

Thus, the expression becomes:

\[
\frac{(x - 1)(x + 1)}{\sqrt{x + 3} - \sqrt{4x}}
\]

### Step 2: Rationalize the denominator
To deal with the square roots in the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{x + 3} + \sqrt{4x} \):

\[
\frac{(x - 1)(x + 1)}{(\sqrt{x + 3} - \sqrt{4x})} \times \frac{\sqrt{x + 3} + \sqrt{4x}}{\sqrt{x + 3} + \sqrt{4x}}
\]

This gives:

\[
\frac{(x - 1)(x + 1)(\sqrt{x + 3} + \sqrt{4x})}{(\sqrt{x + 3})^2 - (\sqrt{4x})^2}
\]

### Step 3: Simplify the denominator
Simplifying the denominator using the difference of squares:

\[
(\sqrt{x + 3})^2 - (\sqrt{4x})^2 = (x + 3) - (4x) = -3x + 3 = 3(1 - x)
\]

Thus, the expression simplifies to:

\[
\frac{(x - 1)(x + 1)(\sqrt{x + 3} + \sqrt{4x})}{3(1 - x)}
\]

### Step 4: Cancel common terms
We can cancel \( x - 1 \) with \( 1 - x \), noting that \( x - 1 = -(1 - x) \). So the expression becomes:

\[
\frac{-(x + 1)(\sqrt{x + 3} + \sqrt{4x})}{3}
\]

### Step 5: Evaluate the limit
Now, substitute \( x = 1 \) into the expression:

\[
\frac{-(1 + 1)(\sqrt{1 + 3} + \sqrt{4 \times 1})}{3} = \frac{-2(\sqrt{4} + \sqrt{4})}{3} = \frac{-2(2 + 2)}{3} = \frac{-8}{3}
\]

Thus, the value of the limit is:

\[
\boxed{-\frac{8}{3}}
\]

Would you like more details on any part of this solution?

### 5 Relative Questions:
1. How is rationalizing the denominator useful in solving limits with square roots?
2. What other types of factoring can be used to simplify limits like this?
3. How do conjugates help simplify expressions involving square roots?
4. Why do we cancel terms before substituting the limit value?
5. What is the significance of the difference of squares in calculus?

### Tip:
When dealing with limits involving square roots, rationalizing the denominator can greatly simplify the expression, making it easier to evaluate the limit.

lim (x → 1) (x² - 1) / (√(x + 3) - √(4x))

Explore how to solve a limit problem involving the expression lim (x → 1) (x² - 1) / (√(x + 3) - √(4x)). This solution uses algebraic techniques like rationalizing the denominator and factoring the difference of squares. A step-by-step solution for calculus students at Grade 12 or college level.

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