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d. begin mathsize 14px style limit as x rightwards arrow 2 of fraction numerator 3 minus square root of 4 x plus 1 end root over denominator x minus 2 end fraction end style  

Pembahasan Soal:

Bentuk begin mathsize 14px style limit as x rightwards arrow a of f open parentheses x close parentheses comma space a not equal to 0 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 2 of fraction numerator 3 minus square root of 4 x plus 1 end root over denominator x minus 2 end fraction end cell equals cell fraction numerator 3 minus square root of 4 open parentheses 2 close parentheses plus 1 end root over denominator 2 minus 2 end fraction end cell row blank equals cell fraction numerator 3 minus square root of 9 over denominator 0 end fraction end cell row blank equals cell fraction numerator 3 minus 3 over denominator 0 end fraction end cell row cell limit as x rightwards arrow 2 of fraction numerator 3 minus square root of 4 x plus 1 end root over denominator x minus 2 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 blank blank cell limit as x rightwards arrow 2 of fraction numerator 3 minus square root of 4 x plus 1 end root over denominator x minus 2 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 3 minus square root of 4 x plus 1 end root over denominator x minus 2 end fraction cross times fraction numerator 3 plus square root of 4 x plus 1 end root over denominator 3 plus square root of 4 x plus 1 end root end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator open parentheses 3 minus square root of 4 x plus 1 end root close parentheses open parentheses 3 plus square root of 4 x plus 1 end root close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses 3 plus square root of 4 x plus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 3 squared minus open parentheses square root of 4 x plus 1 end root close parentheses squared over denominator open parentheses x minus 2 close parentheses open parentheses 3 plus square root of 4 x plus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 9 minus open parentheses 4 x plus 1 close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses 3 plus square root of 4 x plus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator negative 4 x plus 8 over denominator open parentheses x minus 2 close parentheses open parentheses 3 plus square root of 4 x plus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator negative 4 open parentheses x minus 2 close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses 3 plus square root of 4 x plus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator negative 4 over denominator open parentheses 3 plus square root of 4 x plus 1 end root close parentheses end fraction end cell row blank equals cell fraction numerator negative 4 over denominator open parentheses 3 plus square root of 4 open parentheses 2 close parentheses plus 1 end root close parentheses end fraction end cell row blank equals cell fraction numerator negative 4 over denominator 3 plus square root of 9 end fraction end cell row blank equals cell fraction numerator negative 4 over denominator 3 plus 3 end fraction end cell row blank equals cell negative 4 over 6 end cell row blank equals cell negative 2 over 3 end cell end table end style  

Jadi, diperoleh begin mathsize 14px style limit as x rightwards arrow 2 of fraction numerator 3 minus square root of 4 x plus 1 end root over denominator x minus 2 end fraction equals negative 2 over 3 end style.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 13 Maret 2021

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