Pertanyaan

Diketahui x → a lim f ( x ) = − 1 dan x → a lim g ( x ) = 6 . Tentukan nilai: c. x → a lim ( g ( x ) 2 − 6 f ( x ) 2 f ( x ) ⋅ g ( x ) )

Diketahui $x \to a lim f (x) = - 1$ dan $x \to a lim g (x) = 6$ .

Tentukan nilai:

c. $x \to a lim (\frac{f ( x ) \cdot g ( x )}{g ( x ) ^{2} - 6 f ( x ) ^{2}})$

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D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah − 5 1 Ingat! Jika diketahui limit fungsi aljabar x → a lim f ( x ) = L dan x → a lim g ( x ) = M , serta bilangan real , maka: x → a lim [ f ( x ) + g ( x ) ] = x → a lim f ( x ) + x → a lim g ( x ) = L + M x → a lim [ c f ( x ) ] = c ⋅ x → a lim f ( x ) = c ⋅ L x → a lim ( f ( x ) ) n = ( x → a lim f ( x ) ) n = L n x → a lim [ f ( x ) ⋅ g ( x ) ] = x → a lim f ( x ) ⋅ x → a lim g ( x ) = L ⋅ M \ x → a lim [ g ( x ) f ( x ) ] = lim x → a g ( x ) lim x → a f ( x ) = M L = = = = = = lim x → a ( g ( x ) 2 − 6 f ( x ) 2 f ( x ) ⋅ g ( x ) ) l i m x → a g ( x ) 2 − 6 ⋅ f ( x ) 2 l i m x → a f ( x ) ⋅ g ( x ) ( l i m x → a g ( x ) ) 2 − 6 ⋅ ( l i m x → a f ( x ) ) 2 l i m x → a f ( x ) ⋅ l i m x → a g ( x ) 6 2 − 6 ⋅ ( − 1 ) 2 ( − 1 ) ⋅ 6 36 − 6 − 6 30 − 6 5 − 1 Dengan demikian, nilai limit dari x → a lim ( g ( x ) 2 − 6 f ( x ) 2 f ( x ) ⋅ g ( x ) ) , dengan x → a lim f ( x ) = − 1 dan x → a lim g ( x ) = 6 adalah − 5 1 .

Jawaban yang benar untuk pertanyaan tersebut adalah $- \frac{1}{5}$

Ingat!

Jika diketahui limit fungsi aljabar $x \to a lim f (x) = L$ dan $x \to a lim g (x) = M$ , serta bilangan real $c$ , maka:

$x \to a lim [f (x) + g (x)] = x \to a lim f (x) + x \to a lim g (x) = L + M$
$x \to a lim [c f (x)] = c \cdot x \to a lim f (x) = c \cdot L$
$x \to a lim (f (x))^{n} = (x \to a lim f (x))^{n} = L^{n}$
$x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x) = L \cdot M$ \
$x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )} = \frac{L}{M}$

$= = = = = = lim_{x \to a} (\frac{f ( x ) \cdot g ( x )}{g ( x ) ^{2} - 6 f ( x ) ^{2}}) \frac{l i m _{x \to a} f ( x ) \cdot g ( x )}{l i m _{x \to a} g ( x ) ^{2} - 6 \cdot f ( x ) ^{2}} \frac{l i m _{x \to a} f ( x ) \cdot l i m _{x \to a} g ( x )}{( l i m _{x \to a} g ( x ) ) ^{2} - 6 \cdot ( l i m _{x \to a} f ( x ) ) ^{2}} \frac{( - 1 ) \cdot 6}{6 ^{2} - 6 \cdot ( - 1 ) ^{2}} \frac{- 6}{36 - 6} \frac{- 6}{30} \frac{- 1}{5}$

Dengan demikian, nilai limit dari $x \to a lim (\frac{f ( x ) \cdot g ( x )}{g ( x ) ^{2} - 6 f ( x ) ^{2}})$ , dengan $x \to a lim f (x) = - 1$ dan $x \to a lim g (x) = 6$ adalah $- \frac{1}{5}$ .

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