Limits at Infinity, Part I — Question 2

Question 2 - Solution

We are given: $$\lim_{x \to \infty} \left( \sqrt{x^2 + 5x} - x \right)$$

This expression is of the indeterminate form $\infty - \infty$, so we need to rationalize:

Multiply by the conjugate: $$\left( \sqrt{x^2 + 5x} - x \right) \cdot \frac{\sqrt{x^2 + 5x} + x}{\sqrt{x^2 + 5x} + x}$$

This gives: $$\frac{(\sqrt{x^2 + 5x})^2 - x^2}{\sqrt{x^2 + 5x} + x} = \frac{x^2 + 5x - x^2}{\sqrt{x^2 + 5x} + x} = \frac{5x}{\sqrt{x^2 + 5x} + x}$$

Now divide numerator and denominator by $x$: $$\frac{5}{\sqrt{1 + \frac{5}{x}} + 1}$$

As $x \to \infty$, $\frac{5}{x} \to 0$, so: $$\frac{5}{\sqrt{1 + 0} + 1} = \frac{5}{1 + 1} = \frac{5}{2}$$

Final Answer: $$\boxed{\frac{5}{2}}$$