# How do you find vertical, horizontal and oblique asymptotes for #y=1/(2-x)#?

vertical asymptote x = 2

horizontal asymptote y = 0

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation let the denominator equal zero.

solve: 2 - x = 0 → x = 2

When the degree of the numerator < degree of the denominator, as is the case here then the equation is always y=0

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no oblique asymptotes.

Here is the graph of the function. graph{1/(2-x) [-10, 10, -5, 5]}

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Vertical asymptote: Set the denominator equal to zero and solve for x. In this case, ( 2 - x = 0 ) gives ( x = 2 ). Thus, there is a vertical asymptote at ( x = 2 ).

Horizontal asymptote: As x approaches positive or negative infinity, the function approaches a constant value. In this case, as ( x ) goes to infinity or negative infinity, ( y ) approaches ( 0 ) because the degree of the numerator is less than the degree of the denominator. Therefore, there is a horizontal asymptote at ( y = 0 ).

Oblique (slant) asymptote: Check if the degree of the numerator is exactly one greater than the degree of the denominator. If so, perform polynomial long division to find the equation of the oblique asymptote. In this case, since the degree of the numerator is not greater by one, there is no oblique asymptote for the function ( y = \frac{1}{2-x} ).

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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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