You need to force `x^(4.5)` factor out to numerator and `x^(4.6)` to denominator such that:

`lim_(x-gt-oo) (x^(4.5)(3+8/(x^(4.5-1))+sinx/(x^4.5)))/(x^(4.6)(9+7/(x^(4.6-1))+cosx/(x^(4.6))))`

`lim_(x-gt-oo) x^(4.5-4.6)(3+8/(x^(4.5-1))+sinx/((x^4.5)))/(9+7/(x^(4.6-1))+cosx/(x^(4.6)))`

`lim_(x-gt-oo) x^(-0.1)(3+8/(x^(4.5-1))+sinx/(x^4.5))/(9+7/(x^(4.6-1))+cosx/(x^(4.6)))`

`lim_(x-gt-oo) x^(-0.1)*lim_(x-gt-oo)(3+8/(x^(4.5-1))+sinx/(x^4.5))/(9+7/(x^(4.6-1))+cosx/(x^(4.6)))`

`lim_(x-gt-oo) 1/(x^(0.1))*lim_(x-gt-oo)(3+8/(x^(4.5-1))+sinx/(x^4.5))/(9+7/(x^(4.6-1))+cosx/(x^(4.6)))`

`(1/oo)*(3+0+0)/(9+0+0) = 0*3/9 = 0`

**Hence, evaluating the limit of function when x approaches to -`oo ` yields `lim_(x-gt-oo) (3x^(4.5)+8x+sinx)/(9x^(4.6)+7x+cosx) = 0.` **

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