Roboguru

Diberikan x→alim​f(x)=3 dan x→alim​g(x)=−1.  Hitunglah nilai setiap limit berikut. a. x→alim​f2(x)+g2(x)​  b. x→alim​f(x)+g(x)2f(x)−3g(x)​  c. x→alim​3g(x)​[f(x)+3]  d. x→alim​[f(x)−3]4

Pertanyaan

Diberikan limit as x rightwards arrow a of f left parenthesis x right parenthesis equals 3 dan begin mathsize 14px style limit as x rightwards arrow a of g left parenthesis x right parenthesis equals negative 1 end style

Hitunglah nilai setiap limit berikut.

a. begin mathsize 14px style limit as x rightwards arrow a of square root of f squared left parenthesis x right parenthesis plus g squared left parenthesis x right parenthesis end root end style 

b. begin mathsize 14px style limit as x rightwards arrow a of fraction numerator 2 f left parenthesis x right parenthesis minus 3 g left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end fraction end style 

c. begin mathsize 14px style limit as x rightwards arrow a of cube root of g left parenthesis x right parenthesis end root open square brackets f left parenthesis x right parenthesis plus 3 close square brackets end style 

d. begin mathsize 14px style limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis minus 3 close square brackets to the power of 4 end style 

Pembahasan Soal:

Ingat bahwa:

Limit memiliki sifat seperti di bawah ini.

1. 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 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

2. limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis close square brackets equals limit as x rightwards arrow a of f left parenthesis x right parenthesis minus limit as x rightwards arrow a of g left parenthesis x right parenthesis

3. limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis cross times g left parenthesis x right parenthesis close square brackets equals limit as x rightwards arrow a of f left parenthesis x right parenthesis cross times limit as x rightwards arrow a of g left parenthesis x right parenthesis

4. limit as x rightwards arrow a of k f left parenthesis x right parenthesis equals k cross times limit as x rightwards arrow a of f left parenthesis x right parenthesis

5. 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 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

6. limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis close square brackets to the power of n equals open square brackets limit as x rightwards arrow a of f left parenthesis x right parenthesis close square brackets to the power of n

7. limit as x rightwards arrow a of square root of f left parenthesis x right parenthesis end root equals square root of limit as x rightwards arrow a of f left parenthesis x right parenthesis end root

8. limit as x rightwards arrow a of c equals c comma space c element of straight real numbers


Diberikan limit as x rightwards arrow a of f left parenthesis x right parenthesis equals 3 space dan space limit as x rightwards arrow a of g left parenthesis x right parenthesis equals negative 1

a. limit as x rightwards arrow a of square root of f squared left parenthesis x right parenthesis plus g squared left parenthesis x right parenthesis end root

Berdasarkan sifat 1, 6, dan 7 diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow a of square root of f squared left parenthesis x right parenthesis plus g squared left parenthesis x right parenthesis end root end cell equals cell square root of open square brackets limit as x rightwards arrow a of f left parenthesis x right parenthesis close square brackets squared plus open square brackets limit as x rightwards arrow a of g left parenthesis x right parenthesis close square brackets squared end root end cell row blank equals cell square root of open square brackets 3 close square brackets squared plus open square brackets negative 1 close square brackets squared end root end cell row blank equals cell square root of 9 plus 1 end root end cell row blank equals cell square root of 10 end cell end table

Dengan demikian, hasil dari limit as x rightwards arrow a of square root of f squared left parenthesis x right parenthesis plus g squared left parenthesis x right parenthesis end root adalah square root of 10.

b. limit as x rightwards arrow a of fraction numerator 2 f left parenthesis x right parenthesis minus 3 g left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end fraction

Berdasarkan sifat 1, 2, 4, dan 5 diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow a of fraction numerator 2 f left parenthesis x right parenthesis minus 3 g left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end fraction end cell equals cell fraction numerator 2 limit as x rightwards arrow a of f left parenthesis x right parenthesis minus 3 limit as x rightwards arrow a of g left parenthesis x right parenthesis over denominator 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 end fraction end cell row blank equals cell fraction numerator 2 open square brackets 3 close square brackets minus 3 open square brackets negative 1 close square brackets over denominator 3 plus open parentheses negative 1 close parentheses end fraction end cell row blank equals cell fraction numerator 6 plus 3 over denominator 2 end fraction end cell row blank equals cell 9 over 2 end cell end table

Dengan demikian, hasil dari limit as x rightwards arrow a of fraction numerator 2 f left parenthesis x right parenthesis minus 3 g left parenthesis x right parenthesis over denominator f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis end fraction adalah 9 over 2.

c. limit as x rightwards arrow a of cube root of g left parenthesis x right parenthesis end root open square brackets f left parenthesis x right parenthesis plus 3 close square brackets

Berdasarkan sifat 1, 3, 7, dan 8 diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow a of cube root of g left parenthesis x right parenthesis end root open square brackets f left parenthesis x right parenthesis plus 3 close square brackets end cell equals cell cube root of limit as x rightwards arrow a of g left parenthesis x right parenthesis end root cross times open square brackets limit as x rightwards arrow a of f left parenthesis x right parenthesis plus limit as x rightwards arrow a of 3 close square brackets end cell row blank equals cell cube root of negative 1 end root cross times open square brackets 3 plus 3 close square brackets end cell row blank equals cell negative 1 cross times 6 end cell row blank equals cell negative 6 end cell end table

Dengan demikian, hasil dari limit as x rightwards arrow a of cube root of g left parenthesis x right parenthesis end root open square brackets f left parenthesis x right parenthesis plus 3 close square brackets adalah negative 6.

d. limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis minus 3 close square brackets to the power of 4

Berdasarkan sifat 1, 6, dan 8 diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis minus 3 close square brackets to the power of 4 end cell equals cell open square brackets limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis minus 3 close square brackets close square brackets to the power of 4 end cell row blank equals cell open square brackets limit as x rightwards arrow a of f left parenthesis x right parenthesis minus limit as x rightwards arrow a of 3 close square brackets to the power of 4 end cell row blank equals cell open square brackets 3 minus 3 close square brackets to the power of 4 end cell row blank equals cell open square brackets 0 close square brackets to the power of 4 end cell row blank equals 0 end table

Dengan demikian, hasil dari limit as x rightwards arrow a of open square brackets f left parenthesis x right parenthesis minus 3 close square brackets to the power of 4 adalah 0.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

P. Anggrayni

Terakhir diupdate 07 Oktober 2021

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