Question

How do you find the limit of `(xcos(ax))/(tan(bx))` as x approaches 0?

Answer 1

Limit of `(xcos(ax))/tan(bx)`, as `x` approaches `0` is `1/b`

Explanation:

`(xcos(ax))/tan(bx)=(xcos(ax))/(sin(bx)/cos(bx))`

or `(xcos(ax)cos(bx))/sin(bx)`

Hence `L_(x->0)(xcos(ax))/tan(bx)=L_(x->0)(xcos(ax)cos(bx))/sin(bx)`

or `L_(x->0)1/bxx(bx)/sin(bx)xxL_(x->0)cos(ax)xxL_(x->0)cos(bx)`

but as `x->0`, `cospx->1` and `L_(z->0)z/sinz=1`,

the above is equal to

`1/bxxL_(x->0)(bx)/sin(bx)xxL_(x->0)cos(ax)xxL_(x->0)cos(bx)`

or `1/bxx1xx1xx1=1/b`