Computing Limits — Question 8

Question 8

Evaluate the limit: $$\lim_{x \to \infty} \left( \sqrt{x^2 + 5x} - x \right)$$

Question 8 - Solution

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

Step 1: Indeterminate Form

As $x \to \infty$, both $\sqrt{x^2 + 5x}$ and $x$ grow large. The expression becomes $\infty - \infty$, an indeterminate form. We simplify.

Step 2: Multiply by the Conjugate

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

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

Step 3: Divide numerator and denominator by $x$

$$= \frac{5}{\sqrt{1 + \frac{5}{x}} + 1}$$

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

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