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What are the asymptote(s) and hole(s), if any, of # f(x) = lnx/x#?

Answer 1

Nathan Axel

The domain of the function is #D=(0,+oo)# and it's continious in that domain (there are no holes).

We must calculate #lim_{xrarr0^+}f(x)# so let's do it:

#lim_{xrarr0^+}f(x)=lim_{xrarr0^+}lnx/x=lim_{xrarr0^+}1/x*lnx=(+oo)(-oo)=#

#-oo#

Now because #lim_{xrarr0^+}f(x)=-oo#

the line #x=0# is a vertical asymptode of #f(x)#

Now lets calculate #lim_{xrarr+oo}f(x)#

#lim_{xrarr+oo}f(x)=lim_{xrarr+oo}lnx/x#

This limit is #(+oo)/(+oo)# so we can use L'Hôpital's rule, so :

#lim_{xrarr+oo}lnx/x=lim_{xrarr+oo}(1/x)/1=lim_{xrarr+oo}1/x=0#

so the line #y=0# is a horizontal asymptode of #f(x)#

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Answer 2

Isaiah Anthony

The function f(x) = lnx/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. There is no hole in the graph of this function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.