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Diketahui x→alim​f(x)=3, x→alim​h(x)=0, dan x→alim​g(x)=5. Hitunglah nilai setiap limit berikut a. x→alim​[2f(x)+g(x)]2   b. x→alim​4g(x)f(x)⋅h(x)​  c. x→alim​{f3(x)−2g(x)+5h(x)}

Pertanyaan

Diketahui limit as x rightwards arrow a of f open parentheses x close parentheses equals 3begin mathsize 14px style limit as x rightwards arrow a of h left parenthesis x right parenthesis equals 0 end style, dan begin mathsize 14px style limit as x rightwards arrow a of g left parenthesis x right parenthesis equals 5 end style. Hitunglah nilai setiap limit berikut

a. limit as x rightwards arrow a of open square brackets 2 f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis close square brackets squared space  

b. limit as x rightwards arrow a of fraction numerator f left parenthesis x right parenthesis times h left parenthesis x right parenthesis over denominator 4 g left parenthesis x right parenthesis end fraction 

c. limit as x rightwards arrow a of open curly brackets f cubed left parenthesis x right parenthesis minus 2 g left parenthesis x right parenthesis plus 5 h left parenthesis x right parenthesis close curly brackets   

Pembahasan Soal:

Diketahui limit as x rightwards arrow a of f open parentheses x close parentheses equals 3,  begin mathsize 14px style limit as x rightwards arrow a of h left parenthesis x right parenthesis equals 0 end style dan begin mathsize 14px style limit as x rightwards arrow a of g left parenthesis x right parenthesis equals 5 end style

Ingat 

limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis close square brackets equals space limit as x rightwards arrow a of f left parenthesis x right parenthesis plus limit as x rightwards arrow a of g left parenthesis x right parenthesis limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis times g left parenthesis x right parenthesis close square brackets equals space limit as x rightwards arrow a of f left parenthesis x right parenthesis times limit as x rightwards arrow a of g left parenthesis x right parenthesis limit as x rightwards arrow a of open square brackets fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction close square brackets equals space fraction numerator limit as x rightwards arrow a of f left parenthesis x right parenthesis over denominator limit as x rightwards arrow a of g left parenthesis x right parenthesis end fraction comma space limit as x rightwards arrow a of g left parenthesis x right parenthesis not equal to 0 

Perhatikan perhitungan berikut

a. limit as x rightwards arrow a of open square brackets 2 f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis close square brackets squared space

 table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow a of open square brackets 2 f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis close square brackets squared space end cell row blank equals cell limit as x rightwards arrow a of open square brackets 4 f squared left parenthesis x right parenthesis plus 4 f left parenthesis x right parenthesis g left parenthesis x right parenthesis plus g squared left parenthesis x right parenthesis close square brackets space end cell row blank equals cell limit as x rightwards arrow a of 4 f squared left parenthesis x right parenthesis plus limit as x rightwards arrow a of open parentheses 4 f left parenthesis x right parenthesis g left parenthesis x right parenthesis close parentheses plus limit as x rightwards arrow a of g squared left parenthesis x right parenthesis space end cell row blank equals cell 4 f squared left parenthesis a right parenthesis plus 4 limit as x rightwards arrow a of f left parenthesis x right parenthesis times limit as x rightwards arrow a of g left parenthesis x right parenthesis plus 5 squared end cell row blank equals cell 4 f squared left parenthesis a right parenthesis plus 4 f left parenthesis a right parenthesis times 5 plus 25 end cell row blank equals cell 4 f squared left parenthesis a right parenthesis plus 20 f left parenthesis a right parenthesis plus 25 end cell row blank equals cell 4 open parentheses 3 squared close parentheses plus 20 open parentheses 3 close parentheses plus 25 end cell row blank equals cell 4 open parentheses 9 close parentheses plus 60 plus 25 end cell row blank equals 121 end table   

Dengan demikian, nilai dari limit as x rightwards arrow a of open square brackets 2 f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis close square brackets squared space adalah 121.

b. limit as x rightwards arrow a of fraction numerator f left parenthesis x right parenthesis times h left parenthesis x right parenthesis over denominator 4 g left parenthesis x right parenthesis end fraction 

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow a of fraction numerator f left parenthesis x right parenthesis times h left parenthesis x right parenthesis over denominator 4 g left parenthesis x right parenthesis end fraction end cell equals cell fraction numerator limit as x rightwards arrow a of f left parenthesis x right parenthesis times limit as x rightwards arrow a of h left parenthesis x right parenthesis over denominator 4 limit as x rightwards arrow a of g left parenthesis x right parenthesis end fraction end cell row blank equals cell fraction numerator 3 times 0 over denominator 4 times 5 end fraction end cell row blank equals 0 end table  

Dengan demikian, nilai dari limit as x rightwards arrow a of fraction numerator f left parenthesis x right parenthesis times h left parenthesis x right parenthesis over denominator 4 g left parenthesis x right parenthesis end fraction adalah 0.

c. limit as x rightwards arrow a of open curly brackets f cubed left parenthesis x right parenthesis minus 2 g left parenthesis x right parenthesis plus 5 h left parenthesis x right parenthesis close curly brackets

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow a of open curly brackets f cubed left parenthesis x right parenthesis minus 2 g left parenthesis x right parenthesis plus 5 h left parenthesis x right parenthesis close curly brackets end cell row blank equals cell limit as x rightwards arrow a of f cubed left parenthesis x right parenthesis minus 2 limit as x rightwards arrow a of g left parenthesis x right parenthesis plus 5 limit as x rightwards arrow a of h left parenthesis x right parenthesis end cell row blank equals cell 3 cubed minus 2 times 5 plus 5 times 0 end cell row blank equals cell 27 minus 10 end cell row blank equals 17 end table  

Dengan demikian, nilai dari limit as x rightwards arrow a of open curly brackets f cubed left parenthesis x right parenthesis minus 2 g left parenthesis x right parenthesis plus 5 h left parenthesis x right parenthesis close curly brackets adalah 17.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

W. Wati

Terakhir diupdate 07 Oktober 2021

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