x→1lim2+2x−6−2xx3−x2= ....

Pertanyaan

$limit as x rightwards arrow 1 of fraction numerator x cubed minus x squared over denominator square root of 2 plus 2 x end root minus square root of 6 minus 2 x end root end fraction equals$ ....

–2
–1
0
1
2

D. Kamilia

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

i) Rasionalkan penyebut dari bentuk limit di atas

$limit as x rightwards arrow 1 of fraction numerator x cubed minus x squared over denominator square root of 2 plus 2 x end root minus square root of 6 minus 2 x end root end fraction space equals space limit as x rightwards arrow 1 of fraction numerator x cubed minus x squared over denominator square root of 2 plus 2 x end root minus square root of 6 minus 2 x end root end fraction. fraction numerator square root of 2 plus 2 x end root begin display style plus end style square root of 6 minus 2 x end root over denominator square root of 2 plus 2 x end root begin display style plus end style square root of 6 minus 2 x end root end fraction equals space limit as x rightwards arrow 1 of fraction numerator open parentheses x cubed minus x squared close parentheses begin display style open parentheses square root of 2 plus 2 x end root plus square root of 6 minus 2 x end root close parentheses end style over denominator begin display style open parentheses square root of 2 plus 2 x end root close parentheses squared end style begin display style minus end style begin display style begin display style open parentheses square root of 6 minus 2 x end root close parentheses end style squared end style end fraction equals space limit as x rightwards arrow 1 of fraction numerator open parentheses x cubed minus x squared close parentheses begin display style open parentheses square root of 2 plus 2 x end root plus square root of 6 minus 2 x end root close parentheses end style over denominator open parentheses 2 plus 2 x close parentheses begin display style minus end style begin display style open parentheses 6 minus 2 x close parentheses end style end fraction equals space limit as x rightwards arrow 1 of fraction numerator open parentheses x cubed minus x squared close parentheses open parentheses square root of 2 plus 2 x end root plus square root of 6 minus 2 x end root close parentheses over denominator 4 x minus 4 end fraction$

ii) Faktorkan bentuk-bentuk yang dapat difaktorkan

$limit as x rightwards arrow 1 of fraction numerator open parentheses x cubed minus x squared close parentheses open parentheses square root of 2 plus 2 x end root plus square root of 6 minus 2 x end root close parentheses over denominator 4 x minus 4 end fraction space equals space limit as x rightwards arrow 1 of fraction numerator x squared open parentheses x minus 1 close parentheses open parentheses square root of 2 plus 2 x end root plus square root of 6 minus 2 x end root close parentheses over denominator 4 open parentheses x minus 1 close parentheses end fraction equals space limit as x rightwards arrow 1 of fraction numerator begin display style x squared open parentheses square root of 2 plus 2 x end root plus square root of 6 minus 2 x end root close parentheses end style over denominator begin display style 4 end style end fraction$

iii) Substitusikan x = 1 ke dalam bentuk limit

$limit as x rightwards arrow 1 of fraction numerator x squared open parentheses square root of 2 plus 2 x end root plus square root of 6 minus 2 x end root close parentheses over denominator 4 end fraction space equals space fraction numerator begin display style open parentheses 1 close parentheses squared end style begin display style open parentheses square root of 2 plus 2 open parentheses 1 close parentheses end root plus square root of 6 minus 2 open parentheses 1 close parentheses end root close parentheses end style over denominator 4 end fraction space equals space fraction numerator 1 open parentheses square root of 4 plus square root of 4 close parentheses over denominator 4 end fraction equals space 4 over 4 space equals space 1$