# How to find the asymptotes of #f(x) = (x+3) /( x^2 + 8x + 15)# ?

Simplify

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To find the asymptotes of ( f(x) = \frac{x+3}{x^2 + 8x + 15} ), follow these steps:

- Factor the denominator, ( x^2 + 8x + 15 ), to determine any potential vertical asymptotes.
- Identify any horizontal or slant asymptotes by examining the degrees of the numerator and denominator polynomials.

Applying these steps to the given function, ( f(x) = \frac{x+3}{x^2 + 8x + 15} ):

- Factor the denominator: ( x^2 + 8x + 15 = (x + 3)(x + 5) ).
- Determine if there are any vertical asymptotes by setting the denominator equal to zero: ( x^2 + 8x + 15 = 0 ).
- Solving this equation yields no real solutions, indicating no vertical asymptotes.

- Identify horizontal or slant asymptotes by comparing the degrees of the numerator and denominator.
- Since the degree of the numerator (1) is less than the degree of the denominator (2), there is a horizontal asymptote at ( y = 0 ).

Therefore, the function ( f(x) = \frac{x+3}{x^2 + 8x + 15} ) has no vertical asymptotes and a horizontal asymptote at ( y = 0 ).

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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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