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Tentukan nilai limit berikut. c. x→2lim​ x2−6x+8x3−8​

Pertanyaan

Tentukan nilai limit berikut.

c. limit as x rightwards arrow 2 of space fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

nilai dari limit as x rightwards arrow 2 of space fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction adalah negative 6.

Pembahasan

Ingat mengenai konsep limit as x rightwards arrow c of f open parentheses x close parentheses equals f open parentheses c close parentheses.  

Subtitusikan x rightwards arrow 2 ke persamaan fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction, sehingga didapat:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction end cell equals cell fraction numerator 2 cubed minus 8 over denominator 2 squared minus 6 open parentheses 2 close parentheses plus 8 end fraction end cell row blank equals cell fraction numerator 8 minus 8 over denominator 4 minus 12 plus 8 end fraction end cell row blank equals cell 0 over 0 space open parentheses bentuk space tak space tentu close parentheses end cell end table

Karena dengan mensubtitusikan  x rightwards arrow 2 ke persamaan fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction didapat 0 over 0, maka fungsi fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction diubah terlebih dahulu menjadi bentuk yang lebih sederhana dengan cara memfaktorkan pembilang dan penyebut. Ingat kembali mengenai pemfaktoran bentuk a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus a b plus b squared close parentheses.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction end cell equals cell limit as x rightwards arrow 2 of space fraction numerator x cubed minus 2 cubed over denominator x squared minus 6 x plus 8 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator open parentheses x minus 2 close parentheses open parentheses x squared plus 2 x plus 4 close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses x minus 4 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator open parentheses x squared plus 2 x plus 4 close parentheses over denominator open parentheses x minus 4 close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses 2 squared plus 2 open parentheses 2 close parentheses plus 4 close parentheses over denominator open parentheses 2 minus 4 close parentheses end fraction end cell row blank equals cell fraction numerator 4 plus 4 plus 4 over denominator negative 2 end fraction end cell row blank equals cell negative 12 over 2 end cell row blank equals cell negative 6 end cell end table

Dengan demikian, nilai dari limit as x rightwards arrow 2 of space fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction adalah negative 6.

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