x→0lim​x+sin 5xtan 2x+sin x​=...

Pertanyaan

limit as x rightwards arrow 0 of fraction numerator tan space 2 x plus sin space x over denominator x plus sin space 5 x end fraction equals...    

  1.  0

  2.  0 comma 5 

  3.  0 comma 75 

  4.  1  

  5. 2   

P. Anggrayni

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah B.

Pembahasan

Ingat kembali sifat limit dan limit fungsi trigonometri berikut.

  • limit as x rightwards arrow c of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals fraction numerator limit as x rightwards arrow c of space f left parenthesis x right parenthesis over denominator limit as x rightwards arrow c of space g left parenthesis x right parenthesis end fraction
     
  • stack lim space left square bracket with x rightwards arrow c below f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis right square bracket equals limit as x rightwards arrow c of space f left parenthesis x right parenthesis plus limit as x rightwards arrow c of space g left parenthesis x right parenthesis
     
  • limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator b x end fraction equals a over b
     
  • limit as x rightwards arrow 0 of fraction numerator tan space a x over denominator b x end fraction equals a over b

Dari aturan di atas, maka diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator tan space 2 x plus sin space x over denominator x plus sin space 5 x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses tan space 2 x plus sin space x close parentheses over denominator open parentheses x plus sin space 5 x close parentheses end fraction times fraction numerator begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator begin display style fraction numerator tan space 2 x over denominator x end fraction end style plus begin display style fraction numerator sin space x over denominator x end fraction end style over denominator begin display style x over x end style plus begin display style fraction numerator sin space 5 x over denominator x end fraction end style end fraction end cell row blank equals cell fraction numerator limit as x rightwards arrow 0 of space open parentheses begin display style fraction numerator tan space 2 x over denominator x end fraction end style plus begin display style fraction numerator sin space x over denominator x end fraction end style close parentheses over denominator limit as x rightwards arrow 0 of space open parentheses begin display style x over x end style plus begin display style fraction numerator sin space x over denominator x end fraction end style close parentheses end fraction end cell row blank equals cell fraction numerator limit as x rightwards arrow 0 of begin display style fraction numerator tan space 2 x over denominator x end fraction end style plus limit as x rightwards arrow 0 of begin display style fraction numerator sin space x over denominator x end fraction end style over denominator limit as x rightwards arrow 0 of begin display style x over x end style plus limit as x rightwards arrow 0 of begin display style fraction numerator sin space x over denominator x end fraction end style end fraction end cell row blank equals cell fraction numerator 2 plus 1 over denominator 1 plus 5 end fraction end cell row blank equals cell 3 over 6 end cell row blank equals cell 1 half end cell row blank equals cell 0 comma 5 end cell end table     

Dengan demikian, limit as x rightwards arrow 0 of fraction numerator tan space 2 x plus sin space x over denominator x plus sin space 5 x end fraction equals 0 comma 5.

Jadi, jawaban yang benar adalah B.

48

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