limit x->0 (sqrt(1+x)-1)/x — step-by-step solution

1) Let's evaluate the limit of

\frac{\sqrt{1 + x} - 1}{x}

when

x \to 0

The limit of

\sqrt{1 + x}

1

when

x \to 0

The limit of

\sqrt{1 + x} - 1

0

when

x \to 0

For

\frac{\sqrt{1 + x} - 1}{x}

, we encounter the indeterminate form 0/0 as

x

tends to

0

2) We look for a Taylor expansion of the expression:

\sqrt{1 + x}

Apply the Taylor development

\sqrt{A} + \frac{X - A}{2 \sqrt{A}}

of sqrt with X=

1 + x

around A=

1

(order N=1)

\sqrt{1} + \frac{1 + x - 1}{2 \cdot \sqrt{1}}

1 + \frac{0 + x}{2}

3) We redo the calculation using the Taylor expansion:

1 + \frac{x}{2}

\frac{(1 + \frac{x}{2}) - 1}{x}

\frac{\frac{0 \cdot 2 + x}{2}}{x}

\frac{\frac{0 + x}{2}}{x}

\frac{\frac{x}{2}}{x}

\frac{x}{2} \cdot \frac{1}{x}

\frac{1}{2}

So the limit of

\frac{\sqrt{1 + x} - 1}{x}

\frac{1}{2}

when

x \to 0

4) The approximate value of:

\frac{1}{2}

equals:

0.5

▷Result: $\frac{1}{2}$

▷▷Numeric value:

0.5