Limits and Derivatives

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Question

Evaluate $space space space space space space space space space space limit as straight x rightwards arrow 1 of fraction numerator ax squared plus bx plus straight c over denominator cx squared plus bx plus straight a end fraction comma space straight a plus straight b plus straight c space not equal to 0.$

Solution

space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 1 of fraction numerator ax squared plus bc plus straight c over denominator cx squared plus bx plus straight a end fraction equals fraction numerator limit as straight x rightwards arrow 1 of ax squared plus bx plus straight c over denominator limit as straight x rightwards arrow 1 of cx squared plus bx plus straight a end fraction equals fraction numerator straight a left parenthesis 1 right parenthesis squared plus straight b left parenthesis 1 right parenthesis plus straight c over denominator straight c left parenthesis 1 right parenthesis squared plus straight b left parenthesis 1 right parenthesis plus straight a end fraction equals fraction numerator straight a plus straight b plus straight c over denominator straight c plus straight b plus straight a end fraction equals 1

space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight a plus straight b plus straight c not equal to 0 close square brackets