limit x->0 tan(x)/x — step-by-step solution

1) Let's evaluate the limit of

\frac{\tan (x)}{x}

when

x \to 0

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

The limit of

\tan (x)

\tan (0)

when

x \to 0

Let's evaluate:

\tan (0)

ⓘ The tangent is the y-coordinate (vertical axis) divided by the x-coordinate (horizontal axis) of the point reached on the unit circle after a rotation of

0

radians from the horizontal axis pointing to the right, provided the x-coordinate is not zero, this is a remarkable value:

\tan (0)

0

0

For

\frac{\tan (x)}{x}

, we encounter the indeterminate form 0/0 as

x

tends to

0

2) We need a Taylor expansion for:

\tan (x)

Apply the Taylor development

\tan (A) + (1 + {\tan (A)}^{2}) \cdot (X - A)

of tan with X=

x

around A=

0

(order N=1)

\tan (0) + (1 + {\tan (0)}^{2}) \cdot (x + 0)

ⓘ The tangent is the y-coordinate (vertical axis) divided by the x-coordinate (horizontal axis) of the point reached on the unit circle after a rotation of

0

radians from the horizontal axis pointing to the right, provided the x-coordinate is not zero, this is a remarkable value:

\tan (0)

0

■Adding 0 changes nothing:

x + 0

x

0 + (1 + 0^{2}) \cdot x

3) We redo the calculation using the Taylor expansion:

x

\frac{x}{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}{x}

1

1

So the limit of

\frac{\tan (x)}{x}

1

when

x \to 0

▶Limit:

1