Tentukan nilai limit fungsi aljabar berikut
[tex]\displaystyle \lim_{x\to 2}\frac{x^4-3x^3+2x^2-4x+8}{x^4-5x^3+13x^2-24x+20}[/tex]
[tex]\displaystyle \lim_{x\to 2}\frac{x^4-3x^3+2x^2-4x+8}{x^4-5x^3+13x^2-24x+20}[/tex]
[tex] \begin{aligned} \sf \lim_{x\to 2}\frac{x^4-3x^3+2x^2-4x+8}{x^4-5x^3+13x^2-24x+20} &= \sf \lim_{x\to 2} \frac{( {x}^{3} - {x}^{2} - 4)\cancel{(x - 2)}}{( {x}^{3} - {3x}^{2} + 7x - 10) \cancel{(x - 2)}} \\ \sf &=\sf \lim_{x\to 2} \frac{ {x}^{2} (x - 2) + (x - 2)(x + 2) }{ {x}^{2} (x - 2) - x(x - 2) + 5(x - 2)} \\ \sf &=\sf \lim_{x\to 2} \frac{( {x}^{2} + x + 2) \cancel{(x - 2)} }{( {x}^{2} - x + 5) \cancel{(x - 2)}} \\ \sf &= \sf \lim_{x\to 2} \frac{ {x}^{2} + x + 2}{ {x}^{2} - x + 5} \\ \sf &= \sf \frac{ {2}^{2} + 2 + 2 }{ {2}^{2} - 2 + 5 } \\ \sf &= \sf \frac{4 + 2 + 2}{4 - 2 + 5} \\ \sf &=\sf \frac{8}{7} \end{aligned}[/tex]
Verified answer
PEMBAHASAN
Limit 0/0
x⁴ - 5x³ + 13x² - 24x + 20
x⁴ - 3x³ + 2x² - 4x + 8
kurangkan
-2x³ + 11x² - 20x + 20
cek apakah 0/0
subs x = 2
-2.2³ + 11.2² - 20.2 + 12 = -16 + 44 - 28 = 0
limit x→2 ... = 0/0
L'Hopital
karena pemfaktoran agak sulit dilakukan manual untuk pangkat besar (polinomial derajat 4) :)
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[tex]\displaystyle \sf \lim_{x\to 2}\frac{x^4-3x^3+2x^2-4x+8}{x^4-5x^3+13x^2-24x+20} \\ \\ \displaystyle \sf = \lim_{x\to 2}\frac{4x^3-9x^2+4x-4}{4x^3-15x^2+26x-24} \\ \\ \displaystyle \sf = \lim_{x\to 2}\frac{12x^2-18x+4}{12x^2-30x+26} \\ \\ \displaystyle \sf = \frac{6.2^2-9.2+2}{6.2^2-15.2+13} \\ \\ \displaystyle \sf = \boxed{ \frac{8}{7}} [/tex]