What is the limit of sin(6x)/x as x approaches 0?

3 Answers
Jul 23, 2018

lim_(xto0)sin(6x)/x=6

Explanation:

Let ,
L=lim_(xto0)sin(6x)/x=lim_(xto0)sin(6x)/(6x) xx 6

Subst. 6x=theta=>xto 0,then , thetato0

So.

L=lim_(theta to 0) (sintheta)/theta xx 6=(1) xx 6=6

6

Explanation:

\lim_{x\to 0}\frac{\sin(6x)}{x}

=\lim_{x\to 0}\frac{6\sin(6x)}{6x}

=6\lim_{x\to 0}\frac{\sin(6x)}{(6x)}

=6\cdot 1\quad (\because \lim_{t\to 0}\frac{\sin t}{t}=1)

=6

Jul 23, 2018

lim_(x->0)(sin(6x))/x=6

Explanation:

lim_(x->0)(sin(6x))/x

Let y=6x
y/6=x
As x approaches 0, y also approaches 0.

therefore lim_(y->0)(sin(y))/(y/6)
=lim_(y->0)(6sin(y))/y
=6lim_(y->0)sin(y)/y
=6

Note: lim_(a->0)sin(a)/a=1 is a common limit and has been proven countless times.