Pertanyaan

Hitunglah tiap limit fungsi berikut! x → 1 lim 2 − x 2 + 3 1 − x 2

Hitunglah tiap limit fungsi berikut!

$x \to 1 lim \frac{1 - x ^{2}}{2 - x ^{2} + 3}$

S. Amamah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

diperoleh hasil

diperoleh hasil $begin mathsize 14px style lim with italic x yields 1 below fraction numerator 1 minus sign italic x squared over denominator 2 minus sign square root of italic x squared plus 3 end root end fraction equals 4 end style$

Pembahasan

Jika menggunakan metode substitusi maka akan diperoleh nilai limit 0 0 : lim x → 1 2 − x 2 + 3 1 − x 2 = = = = 2 − 1 + 3 1 − 1 2 2 − 4 1 − 1 2 − 2 1 − 1 0 0 Oleh sebab itu, gunakan akar sekawan untuk mengerjakan soal berikut. Akar sekawan dari adalah Dengan demikian, diperoleh hasil

Jika menggunakan metode substitusi maka akan diperoleh nilai limit $\frac{0}{0}$ :

$lim_{x \to 1} \frac{1 - x ^{2}}{2 - x ^{2} + 3} = = = = \frac{1 - 1 ^{2}}{2 - 1 + 3} \frac{1 - 1}{2 - 4} \frac{1 - 1}{2 - 2} \frac{0}{0}$

Oleh sebab itu, gunakan akar sekawan untuk mengerjakan soal berikut. Akar sekawan dari adalah

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator 1 minus x squared over denominator 2 minus square root of x squared plus 3 end root end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator 1 minus x squared over denominator 2 minus square root of x squared plus 3 end root end fraction cross times fraction numerator 2 plus square root of x squared plus 3 end root over denominator 2 plus square root of x squared plus 3 end root end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator left parenthesis 1 minus x squared right parenthesis left parenthesis 2 plus square root of x squared plus 3 end root right parenthesis over denominator 4 plus 2 square root of x squared plus 3 end root minus 2 square root of x squared plus 3 end root minus left parenthesis x squared plus 3 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator left parenthesis 1 minus x squared right parenthesis left parenthesis 2 plus square root of x squared plus 3 end root right parenthesis over denominator 4 minus left parenthesis x squared plus 3 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator open parentheses 1 minus x squared close parentheses open parentheses 2 plus square root of x squared plus 3 end root close parentheses over denominator 4 minus x squared minus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator left parenthesis 1 minus x squared right parenthesis left parenthesis 2 plus square root of x squared plus 3 end root right parenthesis over denominator 1 minus x squared end fraction end cell row blank equals cell limit as x rightwards arrow 1 of 2 plus square root of x squared plus 3 end root end cell row blank equals cell 2 plus square root of 1 squared plus 3 end root end cell row blank equals cell 2 plus square root of 1 plus 3 end root end cell row blank equals cell 2 plus square root of 4 end cell row blank equals cell 2 plus 2 end cell row blank equals 4 end table end style$

Dengan demikian, diperoleh hasil $begin mathsize 14px style lim with italic x yields 1 below fraction numerator 1 minus sign italic x squared over denominator 2 minus sign square root of italic x squared plus 3 end root end fraction equals 4 end style$