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h. begin mathsize 14px style limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator square root of x squared plus 7 end root minus 4 end fraction end style 

Pembahasan Soal:

Bentuk begin mathsize 14px style limit as x rightwards arrow a of f left parenthesis x right parenthesis end style dan begin mathsize 14px style f open parentheses x close parentheses end style merupakan fungsi rasional (pecahan), lakukan tes limit dengan substitusi nilai begin mathsize 14px style x equals a end style ke fungsi undefined, jika diperoleh hasil begin mathsize 14px style 0 over 0 end style, lakukan faktorisasi undefined atas faktor linearnya sehingga penyebab bentuk begin mathsize 14px style 0 over 0 end style, dapat dihilangkan.

Tes limit dengan substitusi nilai begin mathsize 14px style x equals a end style ke fungsi undefined, diperoleh:

  begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator square root of x squared plus 7 end root minus 4 end fraction end cell equals cell fraction numerator open parentheses 3 close parentheses squared minus 9 over denominator square root of open parentheses 3 close parentheses squared plus 7 end root minus 4 end fraction end cell row blank equals cell fraction numerator 9 minus 9 over denominator square root of 16 minus 4 end fraction end cell row blank equals cell fraction numerator 0 over denominator 4 minus 4 end fraction end cell row cell limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator square root of x squared plus 7 end root minus 4 end fraction end cell equals cell 0 over 0 end cell end table end style 

Diperoleh hasil begin mathsize 14px style 0 over 0 end style, sehingga dilakukan faktorisasi undefined dengan mengalikan akar sekawan, perhatikan perhitungan berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator square root of x squared plus 7 end root minus 4 end fraction end cell equals cell limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator square root of x squared plus 7 end root minus 4 end fraction cross times fraction numerator square root of x squared plus 7 end root plus 4 over denominator square root of x squared plus 7 end root plus 4 end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator open parentheses x squared minus 9 close parentheses open parentheses square root of x squared plus 7 end root plus 4 close parentheses over denominator open parentheses square root of x squared plus 7 end root minus 4 close parentheses open parentheses square root of x squared plus 7 end root plus 4 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator left parenthesis x squared minus 9 right parenthesis open parentheses square root of x squared plus 7 end root plus 4 close parentheses over denominator open parentheses square root of x squared plus 7 end root close parentheses squared minus open parentheses 4 close parentheses squared end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator left parenthesis x squared minus 9 right parenthesis open parentheses square root of x squared plus 7 end root plus 4 close parentheses over denominator x squared plus 7 minus 16 end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator left parenthesis x squared minus 9 right parenthesis open parentheses square root of x squared plus 7 end root plus 4 close parentheses over denominator blank end fraction end cell row blank equals cell limit as x rightwards arrow 3 of square root of x squared plus 7 end root plus 4 end cell row blank equals cell square root of open parentheses 3 close parentheses squared plus 7 end root plus 4 end cell row blank equals cell square root of 9 plus 7 end root plus 4 end cell row blank equals cell square root of 16 plus 4 end cell row blank equals cell 4 plus 4 end cell row cell limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator square root of x squared plus 7 end root minus 4 end fraction end cell equals 8 end table end style  

Jadi, diperoleh begin mathsize 14px style limit as x rightwards arrow 3 of fraction numerator x squared minus 9 over denominator square root of x squared plus 7 end root minus 4 end fraction equals 8 end style.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 13 Maret 2021

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