Pertanyaan

Diketahui x → a lim f ( x ) = 3 , x → a lim h ( x ) = 0 , dan x → a lim g ( x ) = 5 . Hitunglah nilai setiap limit berikut a. x → a lim [ 2 f ( x ) + g ( x ) ] 2 b. x → a lim 4 g ( x ) f ( x ) ⋅ h ( x ) c. x → a lim { f 3 ( x ) − 2 g ( x ) + 5 h ( x ) }

Diketahui $x \to a lim f (x) = 3$ , $x \to a lim h (x) = 0$ , dan $x \to a lim g (x) = 5$ . Hitunglah nilai setiap limit berikut

a. $x \to a lim [2 f (x) + g (x)]^{2}$

b. $x \to a lim \frac{f ( x ) \cdot h ( x )}{4 g ( x )}$

c. $x \to a lim {f^{3} (x) - 2 g (x) + 5 h (x)}$

W. Wati

Master Teacher

Jawaban terverifikasi

Jawaban

nilai dari x → a lim { f 3 ( x ) − 2 g ( x ) + 5 h ( x ) } adalah 17 .

nilai dari

x \to a lim {f^{3} (x) - 2 g (x) + 5 h (x)}

adalah

17

Pembahasan

Diketahui x → a lim f ( x ) = 3 , dan Ingat x → a lim [ f ( x ) + g ( x ) ] = x → a lim f ( x ) + x → a lim g ( x ) x → a lim [ f ( x ) ⋅ g ( x ) ] = x → a lim f ( x ) ⋅ x → a lim g ( x ) x → a lim [ g ( x ) f ( x ) ] = lim x → a g ( x ) lim x → a f ( x ) , x → a lim g ( x )  = 0 Perhatikan perhitungan berikut a. x → a lim [ 2 f ( x ) + g ( x ) ] 2 = = = = = = = = lim x → a [ 2 f ( x ) + g ( x ) ] 2 lim x → a [ 4 f 2 ( x ) + 4 f ( x ) g ( x ) + g 2 ( x ) ] lim x → a 4 f 2 ( x ) + lim x → a ( 4 f ( x ) g ( x ) ) + lim x → a g 2 ( x ) 4 f 2 ( a ) + 4 lim x → a f ( x ) ⋅ lim x → a g ( x ) + 5 2 4 f 2 ( a ) + 4 f ( a ) ⋅ 5 + 25 4 f 2 ( a ) + 20 f ( a ) + 25 4 ( 3 2 ) + 20 ( 3 ) + 25 4 ( 9 ) + 60 + 25 121 Dengan demikian, nilai dari x → a lim [ 2 f ( x ) + g ( x ) ] 2 adalah 121. b. x → a lim 4 g ( x ) f ( x ) ⋅ h ( x ) lim x → a 4 g ( x ) f ( x ) ⋅ h ( x ) = = = 4 l i m x → a g ( x ) l i m x → a f ( x ) ⋅ l i m x → a h ( x ) 4 ⋅ 5 3 ⋅ 0 0 Dengan demikian, nilai dari x → a lim 4 g ( x ) f ( x ) ⋅ h ( x ) adalah 0 . c. x → a lim { f 3 ( x ) − 2 g ( x ) + 5 h ( x ) } = = = = lim x → a { f 3 ( x ) − 2 g ( x ) + 5 h ( x ) } lim x → a f 3 ( x ) − 2 lim x → a g ( x ) + 5 lim x → a h ( x ) 3 3 − 2 ⋅ 5 + 5 ⋅ 0 27 − 10 17 Dengan demikian, nilai dari x → a lim { f 3 ( x ) − 2 g ( x ) + 5 h ( x ) } adalah 17 .

Diketahui $x \to a lim f (x) = 3$ , dan

Ingat

$x \to a lim [f (x) + g (x)] = x \to a lim f (x) + x \to a lim g (x) x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x) x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0$

Perhatikan perhitungan berikut

a. $x \to a lim [2 f (x) + g (x)]^{2}$

$= = = = = = = = lim_{x \to a} [2 f (x) + g (x)]^{2} lim_{x \to a} [4 f^{2} (x) + 4 f (x) g (x) + g^{2} (x)] lim_{x \to a} 4 f^{2} (x) + lim_{x \to a} (4 f (x) g (x)) + lim_{x \to a} g^{2} (x) 4 f^{2} (a) + 4 lim_{x \to a} f (x) \cdot lim_{x \to a} g (x) + 5^{2} 4 f^{2} (a) + 4 f (a) \cdot 5 + 25 4 f^{2} (a) + 20 f (a) + 25 4 (3^{2}) + 20 (3) + 25 4 (9) + 60 + 25 121$

Dengan demikian, nilai dari $x \to a lim [2 f (x) + g (x)]^{2}$ adalah 121.

b. $x \to a lim \frac{f ( x ) \cdot h ( x )}{4 g ( x )}$

$lim_{x \to a} \frac{f ( x ) \cdot h ( x )}{4 g ( x )} = = = \frac{l i m _{x \to a} f ( x ) \cdot l i m _{x \to a} h ( x )}{4 l i m _{x \to a} g ( x )} \frac{3 \cdot 0}{4 \cdot 5} 0$

Dengan demikian, nilai dari $x \to a lim \frac{f ( x ) \cdot h ( x )}{4 g ( x )}$ adalah $0$ .

c. $x \to a lim {f^{3} (x) - 2 g (x) + 5 h (x)}$

$= = = = lim_{x \to a} {f^{3} (x) - 2 g (x) + 5 h (x)} lim_{x \to a} f^{3} (x) - 2 lim_{x \to a} g (x) + 5 lim_{x \to a} h (x) 3^{3} - 2 \cdot 5 + 5 \cdot 0 27 - 10 17$

Dengan demikian, nilai dari $x \to a lim {f^{3} (x) - 2 g (x) + 5 h (x)}$ adalah $17$ .

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