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Diketahui x→2lim​f(x)=3 dan nilai limit berikut. (i)  x→2lim​[f(x)−f2(x)]=6(ii)  x→2lim​[(f(x)+2)2−4 f(x)]=13(iii)  x→2lim​[f(x)−2x+4]2=25(iv)  x→2lim​34f2(x)−9​=3(v)   x→2lim​1−f(x)f(x)+9​=−6  Nilai limit yang benar adalah ...

Pertanyaan

Diketahui limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis equals 3 dan nilai limit berikut.

left parenthesis straight i right parenthesis space space limit as straight x rightwards arrow 2 of invisible function application open square brackets straight f left parenthesis straight x right parenthesis minus straight f squared left parenthesis straight x right parenthesis close square brackets equals 6 left parenthesis ii right parenthesis space space limit as straight x rightwards arrow 2 of invisible function application open square brackets open parentheses straight f left parenthesis straight x right parenthesis plus 2 close parentheses squared minus 4 space straight f left parenthesis straight x right parenthesis close square brackets equals 13 left parenthesis iii right parenthesis space space limit as straight x rightwards arrow 2 of invisible function application open square brackets straight f left parenthesis straight x right parenthesis minus 2 straight x plus 4 close square brackets squared equals 25 left parenthesis iv right parenthesis space space limit as straight x rightwards arrow 2 of invisible function application cube root of 4 straight f squared left parenthesis straight x right parenthesis minus 9 end root equals 3 left parenthesis straight v right parenthesis space space space limit as straight x rightwards arrow 2 of invisible function application fraction numerator straight f left parenthesis straight x right parenthesis plus 9 over denominator 1 minus straight f left parenthesis straight x right parenthesis end fraction equals negative 6 

Nilai limit yang benar adalah ...

  1. i, ii, dan iii

  2. i, ii, dan iv

  3. ii, dan iii, dan iv

  4. ii, iv, dan v

  5. iv, dan v

D. Nuryani

Master Teacher

Mahasiswa/Alumni Universitas Padjadjaran

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Ingatlah sifat-sifat limit fungsi untuk menjawab soal di atas.

Akan dicoba satu-satu pernyataan pada soal apakah benar atau tidak. Diketahui limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis equals 3 maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis straight i right parenthesis space space limit as straight x rightwards arrow 2 of invisible function application open square brackets straight f left parenthesis straight x right parenthesis minus straight f squared left parenthesis straight x right parenthesis close square brackets end cell equals cell limit as straight x rightwards arrow 2 of invisible function application f left parenthesis x right parenthesis minus open square brackets limit as straight x rightwards arrow 2 of invisible function application minus straight f left parenthesis straight x right parenthesis close square brackets squared end cell row blank equals cell 3 minus open parentheses 3 close parentheses squared end cell row blank equals cell 3 minus 9 end cell row blank equals cell negative 6. end cell end table

Pernyataan (i) salah.

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis ii right parenthesis space space limit as straight x rightwards arrow 2 of invisible function application open square brackets open parentheses straight f left parenthesis straight x right parenthesis plus 2 close parentheses squared minus 4 space straight f left parenthesis straight x right parenthesis close square brackets end cell equals cell open square brackets space limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis plus 2 close square brackets squared minus 4 limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis end cell row blank equals cell open square brackets space limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis plus space limit as straight x rightwards arrow 2 of invisible function application 2 close square brackets squared minus 4 limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis end cell row blank equals cell open square brackets 3 plus 2 close square brackets squared minus 4 open parentheses 3 close parentheses end cell row blank equals cell 25 minus 12 end cell row blank equals cell 13. end cell end table 

Pernyataan (ii) benar.

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis iii right parenthesis space space limit as straight x rightwards arrow 2 of invisible function application open square brackets straight f left parenthesis straight x right parenthesis minus 2 straight x plus 4 close square brackets squared end cell equals cell space limit as straight x rightwards arrow 2 of invisible function application open square brackets straight f left parenthesis straight x right parenthesis minus 2 straight x plus 4 close square brackets squared end cell row blank equals cell open square brackets limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis minus 2 straight x plus 4 close square brackets squared end cell row blank equals cell open square brackets limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis minus 2 limit as straight x rightwards arrow 2 of invisible function application straight x plus limit as straight x rightwards arrow 2 of invisible function application 4 close square brackets squared end cell row blank equals cell open square brackets 3 minus 2 open parentheses 2 close parentheses plus 4 close square brackets squared end cell row blank equals cell open square brackets 3 minus 4 plus 4 close square brackets squared end cell row blank equals cell 9. end cell end table 

Pertanyaan (iii) salah.

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis iv right parenthesis space space limit as straight x rightwards arrow 2 of invisible function application cube root of 4 straight f squared left parenthesis straight x right parenthesis minus 9 end root end cell equals cell open square brackets limit as straight x rightwards arrow 2 of invisible function application 4 straight f squared left parenthesis straight x right parenthesis minus 9 close square brackets to the power of 1 third end exponent end cell row blank equals cell open square brackets 4 limit as straight x rightwards arrow 2 of invisible function application straight f squared left parenthesis straight x right parenthesis minus limit as straight x rightwards arrow 2 of invisible function application 9 close square brackets to the power of 1 third end exponent end cell row blank equals cell open square brackets 4 open parentheses limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis close parentheses squared minus limit as straight x rightwards arrow 2 of invisible function application 9 close square brackets to the power of 1 third end exponent end cell row blank equals cell open square brackets 4 open parentheses 3 close parentheses squared minus 9 close square brackets to the power of 1 third end exponent end cell row blank equals cell cube root of 36 minus 9 end root end cell row blank equals cell cube root of 27 end cell row blank equals cell 3. end cell end table   

Pertanyaan (iv) benar.

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis straight v right parenthesis space space space limit as straight x rightwards arrow 2 of invisible function application fraction numerator straight f left parenthesis straight x right parenthesis plus 9 over denominator 1 minus straight f left parenthesis straight x right parenthesis end fraction end cell equals cell fraction numerator limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis plus 9 over denominator limit as straight x rightwards arrow 2 of invisible function application 1 minus straight f left parenthesis straight x right parenthesis end fraction end cell row blank equals cell fraction numerator limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis plus limit as straight x rightwards arrow 2 of invisible function application 9 over denominator limit as straight x rightwards arrow 2 of invisible function application 1 minus limit as straight x rightwards arrow 2 of invisible function application straight f left parenthesis straight x right parenthesis end fraction end cell row blank equals cell fraction numerator 3 plus 9 over denominator 1 minus 3 end fraction end cell row blank equals cell fraction numerator 12 over denominator negative 2 end fraction end cell row blank equals cell negative 6. end cell end table 

Pernyataan (v) benar.

Berdasarkan uraian di atas, maka pernyataan yang benar adalah (ii), (iv), dan (v).

Oleh karena itu, jawaban yang benar adalah D.

165

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