# How do you write an equation with a vertical asymptote of #3#, slant asymptote of #y=x+1#, and #x# intercept at #2#?

Let

It has a vertical asymptote at

It has a slant symptote given by

Now, the general polynomial of degree

We know also that

then

so

Finally

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To write an equation with a vertical asymptote of ( x = 3 ), a slant asymptote of ( y = x + 1 ), and an ( x )-intercept at ( x = 2 ), you can use the characteristics provided to construct a rational function.

A rational function that meets these criteria can be expressed in the form:

[ f(x) = \frac{P(x)}{Q(x)} ]

Where ( P(x) ) and ( Q(x) ) are polynomials.

Given the vertical asymptote of ( x = 3 ), we know that ( Q(x) ) should have a factor of ( (x - 3) ) in the denominator to ensure a vertical asymptote at ( x = 3 ).

To satisfy the slant asymptote ( y = x + 1 ), the degree of ( P(x) ) should be one less than the degree of ( Q(x) ), and the leading terms of ( P(x) ) and ( Q(x) ) should represent ( x + 1 ) and ( x - 3 ) respectively.

Finally, since the ( x )-intercept is at ( x = 2 ), we know that ( P(2) = 0 ).

Putting these pieces together, a possible equation for the rational function would be:

[ f(x) = \frac{(x + 1)(x - 2)}{(x - 3)} ]

This function meets all the specified criteria: it has a vertical asymptote at ( x = 3 ), a slant asymptote of ( y = x + 1 ), and an ( x )-intercept at ( x = 2 ).

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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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