How do you find the asymptotes for #y = (x + 1)/(x - 1)#?
vertical asymptote x = 1
horizontal asymptote y = 1
Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.
solve : x - 1 = 0 → x = 1 is the asymptote
Horizontal asymptotes occur as
divide terms on numerator/denominator by x
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To find the asymptotes of the rational function ( y = \frac{x + 1}{x - 1} ), we need to examine the behavior of the function as ( x ) approaches certain values.
- Vertical Asymptotes: Vertical asymptotes occur where the denominator of the rational function equals zero, but the numerator does not. So, we set the denominator equal to zero and solve for ( x ). [ x - 1 = 0 ] [ x = 1 ]
Therefore, there is a vertical asymptote at ( x = 1 ).
- Horizontal Asymptote: To find the horizontal asymptote, we examine the behavior of the function as ( x ) approaches positive or negative infinity. If the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients.
In this case, both the numerator and denominator are of degree 1, so the horizontal asymptote is the ratio of their leading coefficients. [ \text{Horizontal asymptote} = \frac{\text{Coefficient of } x \text{ in numerator}}{\text{Coefficient of } x \text{ in denominator}} ] [ = \frac{1}{1} ] [ = 1 ]
Therefore, the horizontal asymptote is ( y = 1 ).
So, the asymptotes for the given rational function are:
- Vertical asymptote: ( x = 1 )
- Horizontal asymptote: ( y = 1 )
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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.
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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