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What is the limit of # f(x) = (2x+5)/abs(3x-4)# as x goes to infinity?

1 Answer

Oct 18, 2015

#2/3 vv -2/3#, see the explanation.

Explanation:

#|3x-4|=3x-4# for #3x-4>=0, 3x>=4, x>=4/3#

#|3x-4|=-3x+4# for #3x-4<0, 3x<4, x<4/3#

#lim_(x->+oo) (2x+5)/|3x-4| = lim_(x->+oo) (2x+5)/(3x-4) = lim_(x->+oo) (2+5/x)/(3-4/x) = 2/3#

#lim_(x->-oo) (2x+5)/|3x-4| = lim_(x->-oo) (2x+5)/(-3x+4) = lim_(x->-oo) (2+5/x)/(-3+4/x) = -2/3#

Note:
#5/x->0# when #x->oo vv x->-oo#
#4/x->0# when #x->oo vv x->-oo#

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