Determind all horizontal asymptotes of f(x)=((x-2)/(x^2+1))+2

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Determine the horizontal asymptotes of `f(x)=(x-2)/(x^2+1)+2`

Since this is a function, there can be at most two horizontal asymptotes.

(1) To find the asymptote as x increases without bound we can take the limit:

`lim_(x->oo)[(x-2)/(x^2+1)+2]` Dividing numerator and denominator by `x^2` :

`=lim_(x->oo)[(1/x-2/x^2)/(1+1/x^2)+2]`

`=0+2=2`

So the horizontal asymptote as x increases without bound is y=2.

** Note that `1+1/x^2>0;1/x>2/x^2` for x>1 so the function approaches 2 from above**

*** From algebra we know that a rational function with the degree in the numerator is larger than the degree in the denominator goes to 0: adding 2 makes it go to 2 ***

(2) The asymptote as x decreases without bound:

`lim_(x->-oo)[(x-2)/(x^2+1)+2]`

`=lim_(x->-oo)[(1/x-2/x^2)/(1+1/x^2)+2]`

`=0+2=2`

So the horizontal asymptote as x decreases without bound is y=2.

** Here the denominator is positive, but the numerator is negative for x<-1, so the function approaches 2 from below

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The asymptotes to the left and right are both y=2

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The graph:

Sources:

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