Limit x tends to infinity ((x+1)(x+2)(x+3))^1/3-x=?

1 Answer
Jim H
Apr 19, 2018

#1#

Explanation:

#(x+1)(x+2)(x+3) = x^3+bx^2+cx+d#

#root(3)((x+1)(x+2)(x+3)) = root(3)(x^3+bx^2+cx+d)#

# = root(3)(x^3(1+b/x+c/x^2+d/x^3))# #" "# for #x != 0#

# = xroot(3)(1+b/x+c/x^2+d/x^3))# #" "# for #x != 0#

So

#lim_(xrarroo)((x+1)(x+2)(x+3))^(1/3)/x = lim_(xrarroo)root(3)(1+b/x+c/x^2+d/x^3)#

# = root(3)(1+0+0+0) = 1#

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