Evaluate the following limit: `lim_(x->0)(2sin x^0-sin2x^0)/(x^3)`

Answer

1

Explanation

`lim_(x->0)(2sin x^0-sin2x^0)/(x^3)`

`=lim_(x->0)(2sin ((pix)/180)-sin((2xpi)/180))/(x^3)`

`=lim_(x->0)(2sin ((pix)/180)-2sin((xpi)/180)cos((pi x)/180))/(x^3)`

`=lim_(x->0)(2sin ((pix)/180)(1-cos((pix)/180)))/(x^3)`

`=lim_(x->0)(2sin ((pix)/180)(2sin^2((xpi)/360)))/(x^3)`

`=4(lim_(x->0)(sin ((x pi)/180))/x)xx(lim_(x->0)(sin ((x pi)/360))/x)xx(lim_(x->0)(sin ((x pi)/360))/x)`

`=4(lim_(x->0)(sin ((x pi)/180))/x xx pi/180)xx(lim_(x->0)(sin ((x pi)/360))/x xx pi/360)xx(lim_(x->0)(sin ((x pi)/360))/x xx pi/360)`

`=4xx pi/180 xx pi/360 xx pi/360`

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