Pertanyaan

Tentukan nilai limit dari fungsi berikut : 5) lim x → 2 x 2 − 5 x + 6 2 x 2 − 7 x + 3 = ... (faktorisasi)

Tentukan nilai limit dari fungsi berikut :

5) $lim_{x \to 2} \frac{2 x ^{2} - 7 x + 3}{x ^{2} - 5 x + 6} =$ ... (faktorisasi)

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

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$lim subscript x rightwards arrow 2 end subscript space fraction numerator 2 x squared minus 7 x plus 3 over denominator x squared minus 5 x plus 6 end fraction equals negative 1$ .

Pembahasan

Perhatikan perhitungan berikut! Karena hasil limit tersebut ketika disubstitusikan merupakan bilangan tak terdefinisi, maka kita lakukan penyelesaian limit dengan menggunakan pemfaktoran, berikut: Karena hasil limit tersebut ketika difaktorkan juga merupakan bilangan tak terdefinisi, maka kita lakukan penyelesaian limit dengan dalil L'Hopital atau turunan sebagai berikut: Jadi, .

Perhatikan perhitungan berikut!

$lim subscript x rightwards arrow 2 end subscript space fraction numerator 2 x squared minus 7 x plus 3 over denominator x squared minus 5 x plus 6 end fraction equals fraction numerator 2 times 2 squared minus 7 times 2 plus 3 over denominator 2 squared minus 5 times 2 plus 6 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 8 minus 14 plus 3 over denominator 4 minus 10 plus 6 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator negative 3 over denominator 0 end fraction space left parenthesis Tak space terdefinisi right parenthesis$

Karena hasil limit tersebut ketika disubstitusikan x equals 2 merupakan bilangan tak terdefinisi, maka kita lakukan penyelesaian limit dengan menggunakan pemfaktoran, berikut:

$begin mathsize 14px style lim subscript x rightwards arrow 2 end subscript space fraction numerator 2 x squared minus 7 x plus 3 over denominator x squared minus 5 x plus 6 end fraction equals lim subscript x rightwards arrow 2 end subscript space fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses x minus 3 close parentheses over denominator open parentheses x minus 3 close parentheses open parentheses x minus 2 close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals lim subscript x rightwards arrow 2 end subscript fraction numerator 2 x minus 1 over denominator x minus 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 left parenthesis 2 right parenthesis minus 1 over denominator 2 minus 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 3 over 0 space left parenthesis Tak space terdefinisi right parenthesis end style$

Karena hasil limit tersebut ketika difaktorkan juga merupakan bilangan tak terdefinisi, maka kita lakukan penyelesaian limit dengan dalil L'Hopital atau turunan sebagai berikut:

$begin mathsize 14px style lim subscript x rightwards arrow a end subscript space fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals lim subscript x rightwards arrow a end subscript fraction numerator f apostrophe left parenthesis x right parenthesis over denominator g apostrophe left parenthesis x right parenthesis end fraction lim subscript x rightwards arrow 2 end subscript space fraction numerator 2 x squared minus 7 x plus 3 over denominator x squared minus 5 x plus 6 end fraction equals lim subscript x rightwards arrow 2 end subscript fraction numerator 4 x minus 7 over denominator 2 x minus 5 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 4 left parenthesis 2 right parenthesis minus 7 over denominator 2 left parenthesis 2 right parenthesis minus 5 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals negative 1 end style$

Jadi, $lim subscript x rightwards arrow 2 end subscript space fraction numerator 2 x squared minus 7 x plus 3 over denominator x squared minus 5 x plus 6 end fraction equals negative 1$ .