How do you find the vertical, horizontal or slant asymptotes for #h(x) = (2x - 1)/ (6 - x)#?
vertical asymptote x = 6
horizontal asymptote y = -2
Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.
solve : 6 - x = 0 → x = 6 is the equation
If the degree of the numerator and denominator are equal , as they are in this case , both degree 1 . The equation can be found by taking the ratio of leading coefficients.
Here is the graph of the function. graph{(2x-1)/(6-x) [-10, 10, -5, 5]}
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To find the vertical asymptotes of ( h(x) = \frac{2x - 1}{6 - x} ):
Set the denominator equal to zero and solve for ( x ):
[ 6 - x = 0 ] [ x = 6 ]
So, ( x = 6 ) is a vertical asymptote.
To find the horizontal asymptote:
Compare the degrees of the numerator and the denominator:
The degree of the numerator is 1, and the degree of the denominator is also 1.
The horizontal asymptote is given by the ratio of the leading coefficients:
[ y = \frac{2}{-1} ] [ y = -2 ]
So, ( y = -2 ) is the horizontal asymptote.
To check for a slant (oblique) asymptote:
If the degree of the numerator is one more than the degree of the denominator, then there is a slant asymptote.
Here, the degree of the numerator is 1, and the degree of the denominator is also 1. Therefore, there is no slant asymptote for this function.
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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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