Pertanyaan

Tentukanlah x → 2 lim x 2 − 4 x + 4 1 − cos ( x − 2 )

Tentukanlah $x \to 2 lim \frac{1 - cos ( x - 2 )}{x ^{2} - 4 x + 4}$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

.

$limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction equals 1 fourth$ .

Pembahasan

Ingat kembali beberapa sifat berikut. Dengan aturan di atas diperoleh. Jadi, .

Ingat kembali beberapa sifat berikut.

$limit as x rightwards arrow c of fraction numerator sin space a left parenthesis x minus c right parenthesis over denominator b left parenthesis x minus c right parenthesis end fraction equals limit as x rightwards arrow 0 of fraction numerator space a left parenthesis x minus c right parenthesis over denominator sin space b left parenthesis x minus c right parenthesis end fraction equals a over b$

Dengan aturan di atas diperoleh.

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction times fraction numerator 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus cos left parenthesis x minus 2 right parenthesis over denominator left parenthesis x squared minus 4 x plus 4 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus open parentheses 1 minus 2 sin squared begin display style 1 half end style left parenthesis x minus 2 right parenthesis close parentheses over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus 1 plus 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin 1 half left parenthesis x minus 2 right parenthesis times sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis begin display style x minus 2 end style right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell 2 times limit as x rightwards arrow 2 of fraction numerator sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis end fraction times limit as x rightwards arrow 2 of fraction numerator sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis end fraction times limit as x rightwards arrow 2 of fraction numerator 1 over denominator left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell 2 times 1 half times 1 half times fraction numerator 1 over denominator 1 plus square root of cos left parenthesis 2 minus 2 right parenthesis end root end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus square root of cos left parenthesis 0 right parenthesis end root end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus square root of 1 end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus 1 end fraction end cell row blank equals cell 1 half times 1 half end cell row blank equals cell 1 fourth end cell end table$

Jadi, $limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction equals 1 fourth$ .