Limit of $\sqrt{x^{2} + x} - x$ as $x \to \infty$

How to work it out

1. 1

   We want what sqrt($x^{2}$ + x) - x approaches as x → ∞.

   $$\lim_{x \to \infty} \sqrt{x^{2} + x} - x$$
2. 2

   Direct substitution gives an indeterminate form: ∞−∞. It does not tell us the limit.

3. 3

   Method: multiply and divide by √(x²+x) + x, giving x / (√(x²+x) + x) → 1/2.

4. 4

   Result:

   $$\lim_{x \to \infty} \sqrt{x^{2} + x} - x = \frac12$$

Values of sqrt($x^{2}$ + x) - x as x → ∞ (computed)

Questions

More well-known limits

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