Question

Evaluate the following limit. DO NOT USE L'Hospital Rule?

Answer 1

`lim_(xrarr0)sin(12x)/(3x) = 4`

Explanation:

Without using Rules De L'Hospital :)

`lim_(xrarr0)sin(12x)/(3x)`

• `12x=u`

`x=u/12`

`x->0`
`u->0`

`=` `lim_(urarr0)sinu/(cancel(3)*u/cancel(12))` `=` `lim_(urarr0)sinu/(u/(4)` `=` `lim_(urarr0)(sinu/1)/(u/4)` `=`
`=` `lim_(urarr0)4sinu/u` `=` `4lim_(urarr0)sinu/u` `=` `4*1` `=` `4`

Answer 2

A different way to find the limit

Explanation:

We shall use the well-known result `lim_(theta->0) sin theta/theta=1` and the limit constant multiple rule `limaf(x)=alimf(x)`

Also, notice `sin(12x)/(3x)=(4sin(12x))/(4(3x))=4*sin(12x)/(12x)`

`therefore lim_(x->0)sin(12x)/(3x)=4lim_(x->0)sin(12x)/(12x)=4*1=4`