f:( 0; +infinity) R f(x)= -x+2+ ln(x)/x  find asymptotes

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cosinusix | College Teacher | (Level 3) Assistant Educator

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The domain of f is (0,oo) therefore there are no holes in the domain.

At x= 0.

`lim_(x rarr 0)f(x)=2+lim_(x arr 0)ln(x)/x=-oo`


Since the limit is infinite there is a vertical asymptote at x=0

When `x rarr oo`

Recall: if `lim_(x rarr oo) (f(x))/x=a`  and `lim_(x rarr oo) f(x)-ax=b`
then y=ax+b is an oblique asymptote.


In our case, `(f(x))/x=(-x+2+(ln(x))/x)/x=-1+2/x+ln(x)/(x^2)`

`lim_(x rarr oo) (ln x)/(x^2)=0`

Therefore` lim_(x rarr oo) (f(x))/x=-1+0+0`  (a=-1)


` lim_(x rarr oo) f(x)-(-x)=2+lim_(x rarr oo) ln(x)/x=2` (b=2)

y=-x+2 is a oblique asymptote.


Conclusion 2 asymptotes: vertical asymptote x=0 and oblique asymptote y=-x+2