Pertanyaan

x → 8 lim ( x − 8 ) 2 3 x 2 − 4 3 x + 4 = ...

$x \to 8 lim \frac{3 x ^{2} - 4 3 x + 4}{( x - 8 ) ^{2}} = ...$

I. Ridha

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

.

$limit as x rightwards arrow 8 of fraction numerator cube root of x squared end root minus 4 cube root of x plus 4 over denominator open parentheses x minus 8 close parentheses squared end fraction equals 1 over 144$ . space

Pembahasan

Permasalahan limit di atas akan diselesaikan dengan metode pemfaktoran terlebih dahulu karena saat diselesaikanmenggunakan metode subtitusi secara langsung ditemukan hasil berbentuk tak tentu . Sebelum menyelesaikan limit, perlu diketahui salah satu bentuk umum pemfaktoran pangkat dua, yaitu dan bentuk umum pemfaktoran pangkat tiga, yaitu . Denganmenmfaktorkan fungsi pada limit,diperoleh Jadi, .

Permasalahan limit di atas akan diselesaikan dengan metode pemfaktoran terlebih dahulu karena saat diselesaikan menggunakan metode subtitusi secara langsung ditemukan hasil berbentuk tak tentu 0 over 0 .

Sebelum menyelesaikan limit, perlu diketahui salah satu bentuk umum pemfaktoran pangkat dua, yaitu x squared minus 2 x y plus y squared equals open parentheses x minus y close parentheses squared dan bentuk umum pemfaktoran pangkat tiga, yaitu x cubed minus y cubed equals open parentheses x minus y close parentheses open parentheses x squared plus x y plus y squared close parentheses .

Dengan menmfaktorkan fungsi pada limit, diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 8 of fraction numerator cube root of x squared end root minus 4 cube root of x plus 4 over denominator open parentheses x minus 8 close parentheses squared end fraction end cell equals cell limit as x rightwards arrow 8 of open parentheses cube root of x minus 2 close parentheses squared over open parentheses x minus 8 close parentheses squared end cell row blank equals cell open parentheses limit as x rightwards arrow 8 of fraction numerator cube root of x minus 2 over denominator x minus 8 end fraction close parentheses squared end cell row blank equals cell open parentheses limit as x rightwards arrow 8 of fraction numerator cube root of x minus 2 over denominator open parentheses cube root of x close parentheses cubed minus 2 cubed end fraction close parentheses squared end cell row blank equals cell open parentheses limit as x rightwards arrow 8 of fraction numerator cube root of x minus 2 over denominator open parentheses cube root of x minus 2 close parentheses open parentheses open parentheses cube root of x close parentheses squared plus 2 cube root of x plus 4 close parentheses end fraction close parentheses squared end cell row blank equals cell open parentheses limit as x rightwards arrow 8 of fraction numerator up diagonal strike cube root of x minus 2 end strike over denominator up diagonal strike open parentheses cube root of x minus 2 close parentheses end strike open parentheses open parentheses cube root of x close parentheses squared plus 2 cube root of x plus 4 close parentheses end fraction close parentheses squared end cell row blank equals cell open parentheses limit as x rightwards arrow 8 of fraction numerator 1 over denominator open parentheses cube root of x close parentheses squared plus 2 cube root of x plus 4 end fraction close parentheses squared end cell row blank equals cell open parentheses fraction numerator 1 over denominator open parentheses cube root of 8 close parentheses squared plus 2 cube root of 8 plus 4 end fraction close parentheses squared end cell row blank equals cell open parentheses fraction numerator 1 over denominator open parentheses 2 close parentheses squared plus 2 open parentheses 2 close parentheses plus 4 end fraction close parentheses squared end cell row blank equals cell open parentheses 1 over 12 close parentheses squared end cell row blank equals cell 1 over 144 end cell end table$

Jadi, $limit as x rightwards arrow 8 of fraction numerator cube root of x squared end root minus 4 cube root of x plus 4 over denominator open parentheses x minus 8 close parentheses squared end fraction equals 1 over 144$ . space