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 $-6$ .

Pembahasan

Ingat mengenai konsep $\lim_{x\rightarrow c}f\left(x\right)=f\left(c\right)$ .

Subtitusikan $x\rightarrow2$ 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 $x\rightarrow2$ ke persamaan $fraction numerator x cubed minus 8 over denominator x squared minus 6 x plus 8 end fraction$ didapat $\frac00$ , 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 $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$ .

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

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 $-6$ .