Hitunglah nilai limit fungsi berikut:   x→1lim​3x2​−23x​+1(x−1)2​

Pertanyaan

Hitunglah nilai limit fungsi berikut:
 

begin mathsize 14px style limit as x rightwards arrow 1 of fraction numerator open parentheses x minus 1 close parentheses squared over denominator cube root of x squared end root minus 2 cube root of x plus 1 end fraction end style

S. Rahmi

Master Teacher

Jawaban terverifikasi

Jawaban

diperoleh nilai limit as x rightwards arrow 1 of fraction numerator open parentheses x minus 1 close parentheses squared over denominator cube root of x squared end root minus 2 cube root of x plus 1 end fraction equals 9.

Pembahasan

Pembahasan

Untuk mencari nilai limit pada soal dapat dilakukan dengan mensubstitusikan x equals 1 ke bentuk aljabarnya. Perhatikan perhitungan berikut.

limit as x rightwards arrow 1 of fraction numerator open parentheses x minus 1 close parentheses squared over denominator cube root of x squared end root minus 2 cube root of x plus 1 end fraction equals fraction numerator open parentheses 1 minus 1 close parentheses squared over denominator cube root of 1 squared end root minus 2 cube root of 1 plus 1 end fraction equals 0 over 0 

Karena menghasilkan bentuk tak tentu, maka dapat dicari dengan cara lain yaitu pemisalan dan pemfaktoran. 

Misal: begin mathsize 14px style cube root of x equals k left right double arrow k cubed equals x end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis x minus 1 right parenthesis squared end cell equals cell open parentheses k cubed minus 1 close parentheses squared space end cell row blank equals cell open parentheses open parentheses k minus 1 close parentheses open parentheses k squared plus k plus 1 close parentheses close parentheses squared end cell row blank equals cell open parentheses k minus 1 close parentheses squared open parentheses k squared plus k plus 1 close parentheses squared end cell end table end style

Selanjutnya,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as k rightwards arrow 1 of fraction numerator open parentheses k cubed minus 1 close parentheses squared over denominator k squared minus 2 k plus 1 end fraction end cell equals cell limit as k rightwards arrow 1 of fraction numerator open parentheses k minus 1 close parentheses squared open parentheses k squared plus k plus 1 close parentheses squared over denominator open parentheses k minus 1 close parentheses squared end fraction end cell row blank equals cell limit as k rightwards arrow 1 of open parentheses k squared plus k plus 1 close parentheses squared end cell row blank equals 9 end table end style

Dengan demikian, diperoleh nilai limit as x rightwards arrow 1 of fraction numerator open parentheses x minus 1 close parentheses squared over denominator cube root of x squared end root minus 2 cube root of x plus 1 end fraction equals 9.

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