Question

Fin `lim_(x rarr 0) ((cotx)(1-cos2x))/x =2`?

Answer 1

`lim_(x rarr 0) ((cotx)(1-cos2x))/x =2`

Explanation:

We seek:

`L = lim_(x rarr 0) ((cotx)(1-cos2x))/x`
`\ \ = lim_(x rarr 0) ((cosx/sinx)(1-(cos^2x-sin^2x)))/x`
`\ \ = lim_(x rarr 0) (cosx/sinx)((1-(1-sin^2x-sin^2x)))/x`
`\ \ = lim_(x rarr 0) (cosx/sinx)((2sin^2x))/x`
`\ \ = lim_(x rarr 0) (cosx) (2sinx)/x`
`\ \ = 2 lim_(x rarr 0) (cosx) ((sinx)/x)`
`\ \ = 2{ lim_(x rarr 0) (cosx)}{ lim_(x rarr 0) ((sinx)/x)}`
`\ \ = 2 * 1 * 1`
`\ \ = 2`