# How do you find the vertical, horizontal and slant asymptotes of: #(x^2-25)/(x^2+5x)#?

vertical asymptote:

horizontal asymptote:

slant asymptote: does not exist

Finding the Vertical Asymptote

Given,

Factor the numerator.

Cancel out any factors that appear in the numerator and denominator.

Finding the Horizontal Asymptote

Given,

Finding the Slant Asymptote

Given,

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To find the vertical, horizontal, and slant asymptotes of the function ( \frac{x^2 - 25}{x^2 + 5x} ), follow these steps:

**Vertical Asymptotes:**

Vertical asymptotes occur where the denominator of a rational function is equal to zero, provided the numerator is not zero at the same point, as this would result in a hole rather than a vertical asymptote.

Set the denominator ( x^2 + 5x ) equal to zero and solve for ( x ):

[ x^2 + 5x = 0 ] [ x(x + 5) = 0 ]

From this equation, ( x = 0 ) and ( x = -5 ) are the values where the denominator is zero.

**Horizontal Asymptotes:**

To find the horizontal asymptote, compare the degrees of the numerator and the denominator:

- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is ( y = 0 ).
- If the degrees are equal, divide the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator by one, there is a slant (oblique) asymptote.
- If the degree of the numerator is greater than the degree of the denominator by more than one, there is no horizontal asymptote.

Here, both the numerator and denominator have the same degree of 2. So, divide the leading coefficients:

[ \text{Horizontal asymptote} = \frac{1}{1} = 1 ]

So, ( y = 1 ) is the horizontal asymptote.

**Slant Asymptote:**

Since the degrees of the numerator and denominator are the same, there is no slant asymptote.

In summary:

- Vertical asymptotes: ( x = 0 ) and ( x = -5 )
- Horizontal asymptote: ( y = 1 )
- No slant asymptote.

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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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