To find the limit

$\lim_{x \to \infty} \frac{49 - \sqrt{x}}{7 + \sqrt{x}},$

we can start by analyzing the behavior of the numerator and denominator as $x$ approaches infinity.

Step 1: Divide by the highest power of $x$ in the denominator

The highest power of $x$ in the denominator is $\sqrt{x}$, which is $x^{1/2}$. To simplify the expression, we can divide both the numerator and the denominator by $\sqrt{x}$:

$\lim_{x \to \infty} \frac{49 - \sqrt{x}}{7 + \sqrt{x}} = \lim_{x \to \infty} \frac{\frac{49}{\sqrt{x}} - 1}{\frac{7}{\sqrt{x}} + 1}$

Step 2: Simplify the expression

Now, let's take the limit as $x \to \infty$:

• As $x \to \infty$, $\frac{49}{\sqrt{x}} \to 0$ because $\sqrt{x}$ grows without bound.
• Similarly, $\frac{7}{\sqrt{x}} \to 0$ for the same reason.

The expression now simplifies to:

$\lim_{x \to \infty} \frac{0 - 1}{0 + 1} = \frac{-1}{1} = -1$

Conclusion

The limit is

$\lim_{x \to \infty} \frac{49 - \sqrt{x}}{7 + \sqrt{x}} = -1.$

Would you like more details, or do you have any other questions? Here are some related questions you might find interesting:

1. How would the limit change if we replaced $\sqrt{x}$ with $x^n$ where $n > 1/2$?
2. What if the numerator and denominator had different highest powers of $x$? How would we find the limit then?
3. Can we apply L'Hôpital's rule directly to this limit? Why or why not?
4. What are other common techniques for evaluating limits at infinity that involve square roots?
5. How does the behavior of rational functions change when approaching infinity versus approaching zero?

Tip: When evaluating limits involving square roots, dividing by the highest power of $x$ present is often a useful technique to simplify the expression.