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What are the asymptote(s) and hole(s), if any, of #f(x)= x/(2x^3-x+1)#?

Answer 1

Eva Abramson

Asymptote at #x=-1#

No holes.

Factor the denominator:

#f(x)= x/(2x^3-x+1)#

#f(x) = x/((x + 1) (2 x^2 - 2 x + 1))#

If you factor #2 x^2 - 2 x + 1# using the quadratic formula it only has complex roots so the only zero in the denominator is at #x=-1#

Since the factor (x+1) does not cancel the zero is a asymptote not a hole.

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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.

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