Question

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

Answer 1

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

Explanation:

`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]}