Please solve: <span title="lim((Ln(1/x))/(1+Ln(3x)),x,infinity)?">lim((Ln(1/x))/(1+Ln(3x)),...</span>

Question

i know the answer is -1.

i just dont know how to get it.

can anyone show me the work

or explain how it is done?

i totally forgot everything about e and natural logs :/

Answer 1

lim { ln(1/x) / [1 + ln(3x)]} = (- ∞/∞)

x → ∞

you have to recall some properties of logarithms:

log (1/a) = - log a

log (ab) = log a + log b

due to these, your limit becomes:

lim [- lnx / (1 + ln3 + lnx)] =

x → ∞

factor out (lnx) from the denominator:

lim  {- lnx / {lnx [(1/lnx) + (ln3/lnx) + 1]} } =

x → ∞

cancel lnx :

lim  {- 1 / [(1/lnx) + (ln3/lnx) + 1] } =

x → ∞

as you know, lim (x→ ∞) lnx = ∞, so that (1/lnx) and (ln3/lnx) → 0

thus

lim  {- 1 / [(0) + (0) + 1] } = - 1/ 1 = - 1

x → ∞

that agrees with your given answer

I hope it helps...

Bye!