Pertanyaan

x → ∞ lim 25 x 2 − 5 x + 10 − ( 5 x − 3 ) = ...

$x \to \infty lim 25 x^{2} - 5 x + 10 - (5 x - 3) = ...$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

.

limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus open parentheses 5 x minus 3 close parentheses equals 5 over 2

Pembahasan

Cara I Ingat bentuk khusus berikut: Pertama ubah soal ke bentuk khusus, diperoleh Sehingga hasil limit adalah Cara I Jadi, .

Cara I

Ingat bentuk khusus berikut:

$limit as x rightwards arrow infinity of square root of a x squared plus b x plus c end root minus square root of p x squared plus q x plus r end root equals L Jika space a equals p comma space maka space L equals fraction numerator b minus q over denominator 2 square root of a end fraction$

Pertama ubah soal ke bentuk khusus,

diperoleh

a equals p equals 25 b equals negative 5 q equals negative 30

Sehingga hasil limit adalah

$table attributes columnalign right center left columnspacing 0px end attributes row L equals cell fraction numerator b minus q over denominator 2 square root of a end fraction end cell row blank equals cell fraction numerator negative 5 minus left parenthesis negative 30 right parenthesis over denominator 2 square root of 25 end fraction end cell row blank equals cell 25 over 10 end cell row blank equals cell 5 over 2 end cell end table$

Cara I

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus open parentheses 5 x minus 3 close parentheses end cell row blank equals cell limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus square root of open parentheses 5 x minus 3 close parentheses squared end root end cell row blank equals cell limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus square root of 25 x squared minus 30 x plus 9 end root end cell row blank equals cell limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus square root of 25 x squared minus 30 x plus 9 end root cross times fraction numerator square root of 25 x squared minus 5 x plus 10 end root plus square root of 25 x squared minus 30 x plus 9 end root over denominator square root of 25 x squared minus 5 x plus 10 end root plus square root of 25 x squared minus 30 x plus 9 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator left parenthesis 25 x squared minus 5 x plus 10 right parenthesis minus left parenthesis 25 x squared minus 30 x plus 9 right parenthesis over denominator square root of 25 x squared minus 5 x plus 10 end root plus square root of 25 x squared minus 30 x plus 9 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 25 x plus 1 over denominator square root of 25 x squared minus 5 x plus 10 end root plus square root of 25 x squared minus 30 x plus 9 end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style fraction numerator 25 x over denominator x end fraction end style plus begin display style 1 over x end style over denominator square root of begin display style fraction numerator 25 x squared over denominator x squared end fraction end style minus begin display style fraction numerator 5 x over denominator x squared end fraction end style plus begin display style 10 over x squared end style end root plus square root of begin display style fraction numerator 25 x squared over denominator x squared end fraction end style minus begin display style fraction numerator 30 x over denominator x squared end fraction end style plus begin display style 9 over x squared end style end root end fraction end cell row blank blank blank row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 25 plus 1 over x end style over denominator square root of begin display style 25 minus 5 over x plus 10 over x squared end style end root plus square root of begin display style 25 minus 30 over x plus 9 over x squared end style end root end fraction end cell row blank equals cell fraction numerator begin display style 25 plus 0 end style over denominator square root of begin display style 25 minus 0 plus 0 end style end root plus square root of begin display style 25 minus 0 plus 0 end style end root end fraction end cell row blank equals cell fraction numerator begin display style 25 end style over denominator 5 plus 5 end fraction end cell row blank equals cell 25 over 10 end cell row blank equals cell 5 over 2 end cell row blank blank blank end table end style$

Jadi, limit as x rightwards arrow infinity of square root of 25 x squared minus 5 x plus 10 end root minus open parentheses 5 x minus 3 close parentheses equals 5 over 2 .