How do you find vertical, horizontal and oblique asymptotes for #(x^2)/(x-1)#?

Answer 1

Vertical asymptote at #x=1#

Oblique asymptote at #y = x+1#

A vertical asymptote occurs where the denominator is equal to #0#.

So we have:

#x-1=0->x=1#
So a vertical asymptote will occur where #x=1#.

The degree of the numerator is greater than the degree of the denominator so the function will not have horizontal asymptotes but will have oblique ones. To find them: we must split the fraction up like so:

#x^2/(x-1)=x+x/(x-1)=x+1+1/(x-1)#
So for very large values of #x# the fraction term will become insignificantly small and the function will approach the oblique asymptote: #y = x+1#.
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Answer 2

To find the vertical, horizontal, and oblique asymptotes for the function (f(x) = \frac{x^2}{x - 1}), follow these steps:

  1. Vertical Asymptotes: Vertical asymptotes occur where the denominator becomes zero but the numerator does not. Set the denominator equal to zero and solve for (x). In this case, (x - 1 = 0) gives (x = 1). So, there's a vertical asymptote at (x = 1).

  2. Horizontal Asymptotes: To find horizontal asymptotes, examine the behavior of the function as (x) approaches positive or negative infinity. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, there's an oblique asymptote.

  3. Oblique Asymptotes: For oblique asymptotes, perform polynomial long division. Divide (x^2) by (x - 1). The quotient will be the equation of the oblique asymptote. [ \begin{array}{r|l} x + 1 & x \ \hline x^2 - x & \ x^2 - x & -1 \ \hline 0 & 1 \ \end{array} ] So, the oblique asymptote is (y = x + 1).

To summarize:

  • Vertical asymptote: (x = 1)
  • Horizontal asymptote: None
  • Oblique asymptote: (y = x + 1)
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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.

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.

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.

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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