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Limit Fungsi Aljabar ~ Matematika SMA | Catatan Nurkholis

Limit Fungsi Aljabar ~ Matematika SMA

Sifat

Berikut ini beberapa sifat yang harus diketahui mengenai limit fungsi aljabar.

  1. Misalkan $f(a)$ terdefinisi, maka nilai $displaystylelim_{xto a} f(x)=f(a)$
    Contoh:$displaystylelim_{xto6} 5x-11 = 5(6)-11=19$
  2. Jika $displaystyle f(a)=frac{0}{0}$ (tak tentu), maka nilai $displaystylelim_{xto a} f(x)$ diselesaikan dengan operasi aljabar
    • Memfaktorkan pembilang dan penyebut $f(x)$ dengan faktor $(x-a)$ sehingga dapat disederhanakan.
      Contoh:
      $begin{aligned}
      lim_{xto3} frac{x^2-7x+12}{x^2-9}&=lim_{xto3} frac{(x-3)(x-4)}{(x-3)(x+3)}\
      &=lim_{xto3} frac{x-4}{x+3}\
      &=frac{3-4}{3+3}\
      &=-frac{1}{6}end{aligned}$
    • Mengalikan pembilang dan penyebut dengan sekawannya apabila terdapat bentuk akar, kemudian disederhanakan
      Contoh:
      $begin{aligned}
      lim_{xto-2} frac{x+2}{sqrt{5x+14}-2}&=lim_{xto-2}frac{left(x+2right)left(sqrt{5x+14}+2right)}{left(sqrt{5x+14}-2right)left(sqrt{5x+14}+2right)}\
      &=lim_{xto-2}frac{left(x+2right)left(sqrt{5x+14}+2right)}{5x+10}\
      &=lim_{xto-2}frac{left(x+2right)left(sqrt{5x+14}+2right)}{5(x+2)}\
      &=lim_{xto-2}frac{left(sqrt{5x+14}+2right)}{5}\
      &=frac{left(sqrt{5(-2)+14}+2right)}{5}\
      &=frac{4}{5}end{aligned}$

Latihan Soal

Nah, sekarang latihan soal berikut ini bisa dicoba

  1. $displaystylelim_{xto4}frac{x^2-2x-8}{x-4}$
  2. $displaystylelim_{xto16}frac{sqrt{4x}-8}{x-16}$
  3. $displaystylelim_{xto3}frac{x^2-9}{sqrt{10-2x}-(x+1)}$
  4. $displaystylelim_{xto0}frac{sqrt{9+3x}-sqrt{9-3x}}{4x}$

Selamat Belajar!

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