Find the limit as x approaches infinity of (2^x-13^x)/(4^x+12^x)

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You need to evaluate the following limit, such that:

`lim_(x->oo) (2^x-13^x)/(4^x+12^x)`

You need to factor out `13^x` and `12^x` such that:

`lim_(x->oo) (13^x((2/13)^x - 1))/(12^x((4/12)^x+1))`

You need to use the properties of limits, such that:

`lim_(x->oo) (13/12)^x*(lim_(x->oo)(2/13)^x - 1)/(lim_(x->oo)(4/12)^x + 1)`

`lim_(x->oo) (13/12)^x*(0 - 1)/(0 + 1)`

Since `12^x < 13^x => lim_(x->oo) (13/12)^x -> oo`

`lim_(x->oo) (2^x-13^x)/(4^x+12^x) = oo*(-1) = -oo`

Hence, evaluating the given limit yields `lim_(x->oo) (2^x-13^x)/(4^x+12^x) = -oo.`

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