limit as x approaches 0+ of x/(x+|x|)

1) Compute the limit of

\frac{x}{x + | x |}

x \to 0

⁺

ⓘ The function abs is continuous, its limit is its value at the limit of its argument

The limit of

| x |

0

⁺ when

x \to 0

⁺

ⓘ The limit of a sum of finite limits is the sum of the limits

The limit of

x + | x |

0

⁺ when

x \to 0

⁺

For

\frac{x}{x + | x |}

, we encounter the indeterminate form 0/0 as

x

tends to

0

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

| x |

Apply the Taylor development

X

of abs with X=

x

around A=

0

(order N=1)

3) Let's restart using the Taylor expansion:

x

\frac{x}{x + x}

ⓘ Like terms are grouped by adding their coefficients:

x + x

2 x

\frac{x}{2 x}

ⓘ The original denominator must remain nonzero. The fraction is simplified by dividing the numerator and the denominator by their common factor

x

\frac{x}{2 x}

\frac{1}{2}

\frac{1}{2}

So the limit of

\frac{x}{x + | x |}

\frac{1}{2}

when

x \to 0

⁺

▶Limit:

\frac{1}{2}

4) The numeric value of:

\frac{1}{2}

equals:

▷

0.5