Limits and Derivatives

Limits and Derivatives

Question

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

Answer
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 over denominator sin space bx end fraction equals limit as straight x rightwards arrow 0 of open parentheses fraction numerator sin space ax over denominator ax end fraction. fraction numerator bx over denominator sin space bx end fraction. straight a over straight b close parentheses equals straight a over straight b. limit as ax rightwards arrow 0 of fraction numerator sin space ax over denominator ax end fraction. fraction numerator straight a over denominator limit as bx rightwards arrow of space begin display style fraction numerator sin space bx over denominator bx end fraction end style end fraction
                                                                                     [∵ xrightwards arrow0rightwards double arrow ax rightwards arrow0 and bxrightwards arrow0]
space space space space space space space space space space space equals space straight a over straight b minus 1 minus 1 over 1 equals straight a over straight b

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