Hitunglah nilai limit fungsi berikut: x→1lim3x2−23x+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

Untuk mencari nilai limit pada soal dapat dilakukan dengan mensubstitusikan $x=1$ 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:

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