How do you find the asymptotes for # f(x)=x/(x-7)#?
vertical asymptote x = 7 , horizontal asymptote y = 1
Find vertical asymptotes when denominator of a
rational function is zero.
so equate : x - 7 = 0 → x = 7
If the degree of the numerator and the degree of the
denominator are equal a horizontal asymptote exists.
Here they are both of degree 1 and so equal.
To obtain equation take the ratio of coefficients of leading terms.
The graph illustrates these.
graph{x/(x-7) [-10, 10, -5, 5]}
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To find the asymptotes of ( f(x) = \frac{x}{x - 7} ), we identify the vertical and horizontal asymptotes:
-
Vertical asymptote: Set the denominator equal to zero and solve for ( x ): ( x - 7 = 0 ) ( x = 7 ) Therefore, there is a vertical asymptote at ( x = 7 ).
-
Horizontal asymptote: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ). So, the horizontal asymptote for ( f(x) = \frac{x}{x - 7} ) is ( y = 1 ).
Therefore, the vertical asymptote is ( x = 7 ) and the horizontal asymptote is ( 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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