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$limit as x rightwards arrow 1 of space open parentheses fraction numerator 3 over denominator x minus 1 end fraction minus fraction numerator 6 over denominator x squared minus 1 end fraction close parentheses equals$ ...

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$begin mathsize 14px style limit as x rightwards arrow 1 of space open parentheses fraction numerator 3 over denominator x minus 1 end fraction minus fraction numerator 6 over denominator x squared minus 1 end fraction close parentheses equals limit as x rightwards arrow 1 of space open parentheses fraction numerator 3 open parentheses x plus 1 close parentheses minus 6 over denominator open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses end fraction close parentheses 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 limit as x rightwards arrow 1 of open parentheses fraction numerator 3 x minus 3 over denominator open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses end fraction close parentheses 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 limit as x rightwards arrow 1 of open parentheses fraction numerator 3 up diagonal strike open parentheses x minus 1 close parentheses end strike over denominator up diagonal strike open parentheses x minus 1 close parentheses end strike open parentheses x plus 1 close parentheses end fraction close parentheses 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 limit as x rightwards arrow 1 of open parentheses fraction numerator 3 over denominator open parentheses x plus 1 close parentheses end fraction close parentheses 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 fraction numerator 3 over denominator 1 plus 1 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 equals 3 over 2 end style$