limit x->+inf sqrt(x)/x — step-by-step solution

1) Compute the limit of

\frac{\sqrt{x}}{x}

x \to +∞

ⓘ This is a reference limit of the function sqrt

The limit of

\sqrt{x}

+∞

when

x \to +∞

For

\frac{\sqrt{x}}{x}

, we encounter the indeterminate form

\frac{+∞}{+∞}

x

tends to

+∞

2) Let's apply L'Hôpital's rule (limit of a/b = limit of a'/b') by differentiating numerator and denominator:

3) Let's differentiate:

\frac{d}{d x} (\sqrt{x})

ⓘ The derivative of sqrt(x) is 1/(2*sqrt(x)):

\frac{d}{d x} (\sqrt{x})

\frac{1}{2 \sqrt{x}}

\frac{1}{2 \sqrt{x}}

4) Let's differentiate:

\frac{d}{d x} (x)

ⓘ The derivative of x is 1:

\frac{d}{d x} (x)

1

1

Using those derivatives, we obtain a new fraction whose limit can be computed:

ⓘ A quantity that grows without bound multiplied by a quantity not tending to 0 still grows without bound, the signs giving +∞ or -∞

The limit of

2 \sqrt{x}

+∞

when

x \to +∞

ⓘ A number divided by a quantity that grows without bound tends to 0

The limit of

\frac{1}{2 \sqrt{x}}

0

⁺ when

x \to +∞

So the limit of

\frac{\sqrt{x}}{x}

0

⁺ when

x \to +∞

▶Limit:

0