# How to find the asymptotes of #f(x) = 3 / (3 - x)# ?

vertical asymptote x = 3

horizontal asymptote y=0

Vertical asymptotes can be found when the denominator

of a rational function is zero.

hence 3 - x = 0 → x = 3 is a vertical asymptote.

[ If the degree of the numerator is less than the degree of

the denominator of a rational function then y = 0 is a horizontal asymptote ]

here degree of numerator is 0 and degree of denominator 1

hence horizontal asymptote at y = 0

Here is the graph of the function to illustrate them.

graph{3/(3-x) [-10, 10, -5, 5]}

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To find the asymptotes of the function ( f(x) = \frac{3}{3 - x} ), we need to identify any vertical and horizontal asymptotes.

Vertical asymptotes occur where the function approaches positive or negative infinity as ( x ) approaches a certain value. In this case, the function will have a vertical asymptote when the denominator ( 3 - x ) equals zero, leading to division by zero. So, to find the vertical asymptote, we set the denominator equal to zero and solve for ( x ).

( 3 - x = 0 )

( x = 3 )

Hence, the vertical asymptote is at ( x = 3 ).

Horizontal asymptotes occur when the function approaches a constant value as ( x ) approaches positive or negative infinity. To find horizontal asymptotes, we examine the behavior of the function as ( x ) approaches positive or negative infinity.

As ( x ) approaches positive infinity, ( 3 - x ) becomes very large negative, so ( \frac{3}{3 - x} ) approaches zero.

As ( x ) approaches negative infinity, ( 3 - x ) becomes very large positive, so ( \frac{3}{3 - x} ) approaches zero.

Therefore, the horizontal asymptote of the function is ( y = 0 ).

In summary, the function ( f(x) = \frac{3}{3 - x} ) has a vertical asymptote at ( x = 3 ) and a horizontal asymptote at ( y = 0 ).

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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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