Limit of $x^{\frac{1}{x}}$ as $x \to \infty$

How to work it out

1. 1

   We want what $x^{1 / x}$ approaches as x → ∞.

   $$\lim_{x \to \infty} x^{\frac{1}{x}}$$
2. 2

   Direct substitution gives an indeterminate form: ∞^0. It does not tell us the limit.

3. 3

   Method: $x^{1/x}$ = $e^{ln x / x}$ and ln x / x → 0.

4. 4

   Result:

   $$\lim_{x \to \infty} x^{\frac{1}{x}} = 1$$

Values of $x^{1 / x}$ as x → ∞ (computed)

Questions

More well-known limits

• $\left(1 + \frac{1}{x}\right)^{x}$ as $x \to \infty$
• $\left(1 + \frac{2}{x}\right)^{x}$ as $x \to \infty$
• $\left(1 + \frac{3}{x}\right)^{x}$ as $x \to \infty$
• $\left(1 - \frac{1}{x}\right)^{x}$ as $x \to \infty$
• $\frac{\ln\left(x\right)}{x}$ as $x \to \infty$
• $\frac{x}{e^{x}}$ as $x \to \infty$
• $\frac{x^{2}}{e^{x}}$ as $x \to \infty$
• $\frac{\sin\left(x\right)}{x}$ as $x \to \infty$