To find the limit of the expression as $x$ approaches infinity, we start by analyzing the expression:

$\lim_{x \to \infty} \frac{\sqrt{x^2 - 7}}{\sqrt{x - 3}}$

Step 1: Simplify the expression

First, let's simplify the expression by factoring out $x^2$ from the square root in the numerator:

$\sqrt{x^2 - 7} = \sqrt{x^2(1 - \frac{7}{x^2})} = |x|\sqrt{1 - \frac{7}{x^2}}$

Since $x$ is approaching infinity, $|x| = x$ because $x$ is positive. Therefore:

$\sqrt{x^2 - 7} = x\sqrt{1 - \frac{7}{x^2}}$

Now, the limit expression becomes:

$\lim_{x \to \infty} \frac{x\sqrt{1 - \frac{7}{x^2}}}{\sqrt{x - 3}}$

Step 2: Simplify further by dividing both numerator and denominator by $\sqrt{x}$:

$\lim_{x \to \infty} \frac{\sqrt{x}\sqrt{x(1 - \frac{7}{x^2})}}{\sqrt{x}\sqrt{1 - \frac{3}{x}}} = \lim_{x \to \infty} \frac{\sqrt{x(1 - \frac{7}{x^2})}}{\sqrt{1 - \frac{3}{x}}}$

As $x$ approaches infinity, the terms $\frac{7}{x^2}$ and $\frac{3}{x}$ approach 0. Therefore:

$\lim_{x \to \infty} \frac{\sqrt{x(1 - 0)}}{\sqrt{1 - 0}} = \lim_{x \to \infty} \sqrt{x} = \infty$

Final Answer:

The limit is:

$\lim_{x \to \infty} \frac{\sqrt{x^2 - 7}}{\sqrt{x - 3}} = \infty$

This means the expression grows without bound as $x$ approaches infinity.

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Would you like any details on the steps? If not, here are some follow-up questions:

1. What happens if $x$ approaches negative infinity instead?
2. How would the result change if the numerator was $\sqrt{x^2 + 7}$?
3. What is the limit of the same expression as $x$ approaches zero?
4. Can you find the limit for a similar expression: $\lim_{x \to \infty} \frac{\sqrt{x^2 - 1}}{\sqrt{x + 1}}$?
5. How does the limit change if the expression is squared?

Tip: When dealing with square roots involving large $x$, it's useful to factor out the dominant term to simplify the expression.