What is #lim_(xrarroo)(ln(9x+10)-ln(2+3x^2))#?

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Answer 1 · 2015-11-08T20:44:04

#lim_(xrarroo)(ln(9x+10)-ln(2+3x^2)) = -oo#

#lim_(xrarroo)(ln(9x+10)-ln(2+3x^2)) = lim_(xrarroo)ln((9x+10)/(3x^2+2))#

Note that #lim_(xrarroo)((9x+10)/(3x^2+2)) = 0#.

And recall that #lim_(urarr0^+)lnu = -oo#, so

#lim_(xrarroo)(ln(9x+10)-ln(2+3x^2)) = lim_(xrarroo)ln((9x+10)/(3x^2+2)) = -oo#

Note

The function goes to #-oo# as #xrarroo#, but it goes very, very slowly. Here is the graph:

graph{(ln(9x+10)-ln(2+3x^2)) [-23.46, 41.47, -21.23, 11.23]}