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

Let's evaluate the limit of

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

when

x \to 0

The limit of

\sin (x)

0

when

x \to 0

For

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

, we encounter the indeterminate form 0/0 as

x

tends to

0

Let's apply L'Hôpital's rule (limit of a/b = limit of a'/b') by differentiating numerator and denominator:

Let's perform the differentiation:

{(\sin (x))}^{'}

\cos (x)

Let's perform the differentiation:

{(x)}^{'}

1

Using those derivatives, we obtain a new fraction whose limit can be computed:

The limit of

\cos (x)

1

when

x \to 0

So the limit of

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

1

when

x \to 0

▷Result: $1$