Wyznacz granicę ciągu o wyrazie ogólnym:
(x - 1)^4 = (x - 1)² * (x - 1)² = (x² - 2x + 1)(x² - 2x + 1) =
x^4 - 2x³ + x² - 2x³ + 4x² - 2x + x² - 2x + 1 = x^4 - 4x³ + 6x² - 4x + 1
lim an = lim [ ( x^4 - 4x³ + 6x² - 4x + 1 ) / (2x^4 + 8)] =
lim [ ( x^4/x^4 - 4x³/x^4 + 6x²/x^4 - 4x/x^4 + 1/x^4 ) / (2x^4/x^4 + 8/x^4)] =
lim [ ( 1 - 4/x + 6/x^2 - 4/x^3 + 1/x^4 ) / (2 + 8/x^4)] =
[( 1 - 0 + 0 - 0 + 0) / (2 + 0) ] = 1 / 2
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(x - 1)^4 = (x - 1)² * (x - 1)² = (x² - 2x + 1)(x² - 2x + 1) =
x^4 - 2x³ + x² - 2x³ + 4x² - 2x + x² - 2x + 1 = x^4 - 4x³ + 6x² - 4x + 1
lim an = lim [ ( x^4 - 4x³ + 6x² - 4x + 1 ) / (2x^4 + 8)] =
lim [ ( x^4/x^4 - 4x³/x^4 + 6x²/x^4 - 4x/x^4 + 1/x^4 ) / (2x^4/x^4 + 8/x^4)] =
lim [ ( 1 - 4/x + 6/x^2 - 4/x^3 + 1/x^4 ) / (2 + 8/x^4)] =
[( 1 - 0 + 0 - 0 + 0) / (2 + 0) ] = 1 / 2