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

Compute the limit of

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

x \to 0

The limit of

\tan (x)

0

when

x \to 0

For

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

, the limit takes the indeterminate form 0/0 when

x

→

0

We look for a Taylor expansion of the expression:

\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)

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

Let's restart using the Taylor expansion:

x

\frac{x}{x}

1

So the limit of

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

1

when

x \to 0

▷Answer: $1$