Limits and Derivatives

Limits and Derivatives

Question

Evaluate space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space space ax plus bx over denominator ax plus sin space bx end fraction comma space straight a comma space space straight b comma space straight a plus straight b not equal to 0

Answer
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 0 of fraction numerator sin space ax plus bx over denominator ax plus sin space bx end fraction equals limit as straight x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin space ax space plus space space bx over denominator straight x end fraction end style over denominator begin display style fraction numerator ax plus sin space space bx over denominator straight x end fraction end style end fraction equals fraction numerator begin display style fraction numerator sin space ax over denominator straight x end fraction plus straight b end style over denominator straight a plus begin display style fraction numerator sin space bx over denominator straight x end fraction end style end fraction

    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 space space space space space space space space space space space space space space space space space equals space limit as straight x rightwards arrow 0 of fraction numerator begin display style fraction numerator straight a space sin space ax over denominator ax end fraction plus straight b end style over denominator straight a plus begin display style fraction numerator straight b space sin space bx over denominator bx end fraction end style end fraction equals fraction numerator straight a. limit as ax rightwards arrow 0 of begin display style fraction numerator siin space ax over denominator ax end fraction end style plus straight b over denominator straight a plus straight b. limit as bx rightwards arrow 0 of begin display style fraction numerator sin space bx over denominator bx end fraction end style end fraction equals fraction numerator straight a plus straight b over denominator straight a plus straight b end fraction equals 1

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