If y = 3 is a horizontal asymptote of a rational function, which must be true?

lim
x→ 3
f(x) = 0 <<my answer

lim
x→ 0
f(x) = 3

lim
x→ ∞
f(x) = 3

lim
x→ 3
f(x) = ∞

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3 as an asymptote means that is the value that the function approaches as x goes to infinity

To determine the choices that must be true if y = 3 is a horizontal asymptote of a rational function, let's first understand what a horizontal asymptote represents.

A horizontal asymptote is a line that the function approaches as x becomes extremely large or extremely small. In the case of a rational function, it is determined by the degrees of the numerator and denominator.

Now, let's consider the given options:

Option 1: lim(x→3) f(x) = 0
This option does not necessarily have to be true. A horizontal asymptote of y = 3 means that the function approaches 3 as x becomes infinitely large, not necessarily when x approaches a specific value like 3.

Option 2: lim(x→0) f(x) = 3
This option does not have to be true either. A horizontal asymptote at y = 3 does not indicate the behavior of the function at x = 0.

Option 3: lim(x→∞) f(x) = 3
This option must be true if y = 3 is a horizontal asymptote. It indicates that as x becomes extremely large (approaches infinity), the function approaches the horizontal line y = 3.

Option 4: lim(x→3) f(x) = ∞
This option does not necessarily have to be true. A horizontal asymptote at y = 3 does not suggest that the function approaches infinity as x approaches 3.

Therefore, the correct answer is:

lim(x→∞) f(x) = 3