# How do you find the vertical, horizontal or slant asymptotes for #(4x^2+5)/( x^2-1)#?

Vertical asymptotes:

Horizontal asymptote:

I'm not sure how complicated your issues will get, but at this point, either a removable discontinuity or a vertical asymptote could occur. If you don't know what a removable discontinuity is, you can probably skip the following section.

Discontinuities that can be removed:

Try factoring both the top and the bottom to see if you have a removable discontinuity; if any of the factors cancel, that's when you have a removable discontinuity. Here's an example:

This contributes to:

Final Breakable Interruptions

Let's move on to the asymptotes that are horizontal and oblique, or slant as you put it.

If the top is larger, simplify! Your oblique asymptote is what you end up with.

I hope this was useful.

Moore, Jonathan 'JMoney'

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To find the vertical asymptotes of the function (4x^2 + 5)/(x^2 - 1), we identify the values of x where the denominator equals zero, excluding any roots that cancel out with the numerator.

To find horizontal asymptotes, we compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at y = 0. If the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

To find slant (oblique) asymptotes, we perform polynomial division to see if the function can be written as a polynomial plus a proper rational function. If the degree of the polynomial obtained from the division is one greater than the degree of the denominator, there is a slant asymptote.

To summarize:

- Vertical asymptotes: Set the denominator equal to zero and solve for x, excluding any roots that cancel out with the numerator.
- Horizontal asymptotes: Compare the degrees of the numerator and denominator.
- Slant asymptotes: Perform polynomial division to see if the function can be expressed as a polynomial plus a proper rational function.

For the given function (4x^2 + 5)/(x^2 - 1):

- Vertical asymptotes: Set x^2 - 1 = 0 ⇒ x^2 = 1 ⇒ x = ±1.
- Horizontal asymptotes: The degrees of the numerator and denominator are equal, so we look at the ratio of leading coefficients: 4/1 = 4. Therefore, there is a horizontal asymptote at y = 4.
- There is no slant asymptote because the degree of the numerator (2) is not one greater than the degree of the denominator (2).

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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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