Limit of $\sqrt{x + 1} - \sqrt{x}$ as $x \to \infty$

How to work it out

1. 1

   We want what sqrt(x + 1) - sqrt(x) approaches as x → ∞.

   $$\lim_{x \to \infty} \sqrt{x + 1} - \sqrt{x}$$
2. 2

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

3. 3

   Method: multiply by the conjugate: 1 / (√(x+1) + √x) → 0.

4. 4

   Result:

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

Values of sqrt(x + 1) - sqrt(x) as x → ∞ (computed)

Questions

More well-known limits

• $\left(1 + \frac{1}{x}\right)^{x}$ as $x \to \infty$
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