# How do you find the vertical, horizontal or slant asymptotes for #f(x) = (1/x) + 3#?

Horizontal:

Vertical :

Tn the quadratic form,

Likewise,

So,

the vertical asy,ptote is x = 0 and

the horizontal asymptote is y = 3.#.

Note

hyperbola having the guiding asymptotes

Here, from (1), it is immediate that

x(y-3)=0 gives the pair of perpendicular asymptotes.

See the Socratic graph.

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To find the vertical asymptote(s) of the function ( f(x) = \frac{1}{x} + 3 ), observe where the denominator becomes zero, as division by zero is undefined. In this case, the vertical asymptote occurs when ( x = 0 ) since the denominator ( x ) approaches zero as ( x ) approaches zero.

To find horizontal asymptotes, examine the behavior of the function as ( x ) approaches positive or negative infinity. As ( x ) becomes very large (positive or negative), ( \frac{1}{x} ) approaches zero. Therefore, the horizontal asymptote is ( y = 3 ).

There are no slant asymptotes for this function since the degree of the numerator is less than the degree of the denominator.

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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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