# How do you find the asymptotes for #f(x)= (4x^2+5)/( x^2-1)#?

To find vertical asymptote, you must set the denominator to 0 and then solve.

Now for horizontal asymptotes, which are a little trickier.

Verification:

This proves that there is a horizontal asymptote at y = 4, because an asymptote is essentially and undefined line on the graph of a function.

Practice exercises:

Good luck!

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To find the asymptotes of ( f(x) = \frac{4x^2 + 5}{x^2 - 1} ), we need to examine the behavior of the function as ( x ) approaches certain values.

- Vertical asymptotes occur where the denominator of the rational function equals zero but the numerator does not. So, set the denominator ( x^2 - 1 ) equal to zero and solve for ( x ):

[ x^2 - 1 = 0 ] [ (x - 1)(x + 1) = 0 ] [ x = 1 \quad \text{or} \quad x = -1 ]

Therefore, the vertical asymptotes are ( x = 1 ) and ( x = -1 ).

- Horizontal asymptotes can be found by analyzing the behavior of the function as ( x ) approaches positive or negative infinity. To do this, divide the leading terms of the numerator and denominator:

[ \lim_{{x \to \infty}} \frac{4x^2}{x^2} = 4 ] [ \lim_{{x \to -\infty}} \frac{4x^2}{x^2} = 4 ]

Since the limit as ( x ) approaches infinity or negative infinity is a constant (4), the horizontal asymptote is ( y = 4 ).

Therefore, the asymptotes for the function ( f(x) = \frac{4x^2 + 5}{x^2 - 1} ) are:

- Vertical asymptotes at ( x = 1 ) and ( x = -1 )
- Horizontal asymptote at ( y = 4 ).

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To find the asymptotes of ( f(x) = \frac{4x^2 + 5}{x^2 - 1} ), you need to consider both vertical and horizontal asymptotes.

Vertical asymptotes occur where the denominator of the function becomes zero but the numerator does not. In this case, the denominator (x^2 - 1) becomes zero when (x = 1) or (x = -1). Therefore, the vertical asymptotes are (x = 1) and (x = -1).

Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator. In this case, since both the numerator and the denominator are quadratic, we can compare the leading coefficients. The horizontal asymptote can be found by dividing the leading term of the numerator by the leading term of the denominator. Here, the horizontal asymptote is (y = 4).

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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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