limit x->0 sin(2x)/x — step-by-step solution | Explain MAths

limit x->0 sin(2x)/x

Let's evaluate the limit of

\frac{\sin (2 x)}{x}

when

x \to 0

The limit of

2 x

is

0

when

x \to 0

The limit of

\sin (2 x)

is

0

when

x \to 0

For

\frac{\sin (2 x)}{x}

, the limit takes the indeterminate form 0/0 when

x

→

0

We look for a Taylor expansion of the expression:

\sin (2 x)

Apply the Taylor development

\sin (A) + \cos (A) \cdot (X - A)

of sin with X=

2 x

around A=

0

(order N=1)

\sin (0) + \cos (0) \cdot (2 x + 0)

0 + 2 x

Let's restart using the Taylor expansion:

2 x

\frac{2 x}{x}

2

So the limit of

\frac{\sin (2 x)}{x}

is

2

when

x \to 0

▷Result: $2$