Question 1086385
a)  Lim x->infinity { x - ln(x) }  = Lim x->infinity { {{{ ln(e^x / x) }}} }

Let {{{ u = e^x / x }}}   (i.e. just look at the inside of ln() )

Lim x->infinity {u}  = Lim u->infinity {u}   ( letting x -> infinity is equivalent to letting u->infinity)

Lim u->infinity {{{ (ln(u)) }}}  = infinity

Ans:  The limit does not exist, it goes off to infinity.

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b)   |{{{x^3+x^2-6x) }}}| = | {{{ x(x^2+x-6) }}} | = |{{{ x(x-2)(x+3) }}} |

Roots at  x=-3, x=0, and x=2.   The function is always >= 0.   Graph below


*[illustration graph]