Question

Wyznacz granicę ciągu o wyrazie ogólnym:

Answer 1

(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