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Jika x → 0 lim ​ f ( x ) = − 5 dan x → 0 lim ​ g ( x ) = 1 , maka x → 0 lim ​ ( g 2 ( x ) f 2 ( x ) ​ + f ( x ) g ( x ) ​ + 9 ​ ) = ...

Jika  dan , maka 

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A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

nilai adalah .

nilai begin mathsize 14px style limit as x rightwards arrow 0 of open parentheses fraction numerator f squared open parentheses x close parentheses over denominator g squared open parentheses x close parentheses end fraction plus fraction numerator square root of g open parentheses x close parentheses end root plus 9 over denominator f open parentheses x close parentheses end fraction close parentheses end style adalah begin mathsize 14px style 23 end style.

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Pembahasan

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Perhatikan perhitungan berikut. Ingat, sifat limit: Maka: Jadi, nilai adalah .

Perhatikan perhitungan berikut.

Ingat, sifat limit:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow c of open parentheses f open parentheses x close parentheses plus-or-minus g open parentheses x close parentheses close parentheses end cell equals cell limit as x rightwards arrow c of f open parentheses x close parentheses plus limit as x rightwards arrow c of g open parentheses x close parentheses end cell row cell limit as x rightwards arrow c of open parentheses fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction close parentheses end cell equals cell fraction numerator limit as x rightwards arrow c of f open parentheses x close parentheses over denominator limit as x rightwards arrow c of g open parentheses x close parentheses end fraction comma space limit as x rightwards arrow c of g open parentheses x close parentheses not equal to 0 end cell row cell limit as x rightwards arrow c of open square brackets f open parentheses x close parentheses close square brackets to the power of n end cell equals cell open square brackets limit as x rightwards arrow c of f open parentheses x close parentheses close square brackets to the power of n end cell row cell limit as x rightwards arrow c of a end cell equals a end table end style

Maka:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of open parentheses fraction numerator f squared open parentheses x close parentheses over denominator g squared open parentheses x close parentheses end fraction plus fraction numerator square root of g open parentheses x close parentheses end root plus 9 over denominator f open parentheses x close parentheses end fraction close parentheses end cell equals cell limit as x rightwards arrow 0 of open parentheses fraction numerator f squared open parentheses x close parentheses over denominator g squared open parentheses x close parentheses end fraction close parentheses plus limit as x rightwards arrow 0 of open parentheses fraction numerator square root of g open parentheses x close parentheses end root plus 9 over denominator f open parentheses x close parentheses end fraction close parentheses end cell row blank equals cell fraction numerator limit as x rightwards arrow 0 of open parentheses f squared left parenthesis x right parenthesis close parentheses over denominator limit as x rightwards arrow 0 of left parenthesis g squared left parenthesis x right parenthesis right parenthesis end fraction plus fraction numerator limit as x rightwards arrow 0 of open parentheses square root of g left parenthesis x right parenthesis end root plus 9 close parentheses over denominator limit as x rightwards arrow 0 of left parenthesis f left parenthesis x right parenthesis right parenthesis end fraction end cell row blank equals cell fraction numerator limit as x rightwards arrow 0 of left parenthesis f squared left parenthesis x right parenthesis right parenthesis over denominator limit as x rightwards arrow 0 of left parenthesis g squared left parenthesis x right parenthesis right parenthesis end fraction plus fraction numerator limit as x rightwards arrow 0 of open parentheses square root of g left parenthesis x right parenthesis end root close parentheses plus limit as x rightwards arrow 0 of 9 over denominator limit as x rightwards arrow 0 of left parenthesis f left parenthesis x right parenthesis right parenthesis end fraction end cell row blank equals cell open square brackets limit as x rightwards arrow 0 of left parenthesis f left parenthesis x right parenthesis right parenthesis close square brackets squared over open square brackets limit as x rightwards arrow 0 of left parenthesis g left parenthesis x right parenthesis right parenthesis close square brackets squared plus fraction numerator square root of limit as x rightwards arrow 0 of open parentheses g left parenthesis x right parenthesis close parentheses end root plus limit as x rightwards arrow 0 of 9 over denominator limit as x rightwards arrow 0 of left parenthesis f left parenthesis x right parenthesis right parenthesis end fraction end cell row blank equals cell open parentheses negative 5 close parentheses squared over 1 squared plus fraction numerator square root of 1 plus 9 over denominator negative 5 end fraction end cell row blank equals cell 25 over 1 plus fraction numerator 1 plus 9 over denominator negative 5 end fraction end cell row blank equals cell 25 minus 2 end cell row blank equals 23 end table end style

Jadi, nilai begin mathsize 14px style limit as x rightwards arrow 0 of open parentheses fraction numerator f squared open parentheses x close parentheses over denominator g squared open parentheses x close parentheses end fraction plus fraction numerator square root of g open parentheses x close parentheses end root plus 9 over denominator f open parentheses x close parentheses end fraction close parentheses end style adalah begin mathsize 14px style 23 end style.

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Perhatikan sifat-sifat limit berikut: I. x → c lim ​ x 2 + x = x → c lim ​ x 2 + x → c lim ​ x II. x → c lim ​ x + 1 ​ = x → c lim ​ x ​ + x → c lim ​ 1 III. x → c lim ​ ( 4 x + 1 ) 3 = ( x →...

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