If y = 3 is a horizontal asymptote of a rational function, which must be true?
A)
lim
x→ 3
f(x) = 0
B)
lim
x→ 0
f(x) = 3
C)
lim
x→ ∞
f(x) = 3
D)
lim
x→ 3
f(x) = ∞
My answer is B
no, that would make it seem like y(0) = 3
But that is not an asymptote. You get a horizontal asymptote when y approaches some value for very large x. Try (C)
http://www.wolframalpha.com/input/?i=plot+y%3D3+%2B+1%2Fx,+y%3D3-3%2F(x%5E2%2B1)
To determine which statement is true based on the given information, we need to understand the concept of horizontal asymptotes for rational functions.
A horizontal asymptote is a line that the graph of a function approaches as the input approaches positive or negative infinity. In this case, we are given that y = 3 is a horizontal asymptote of a rational function.
The horizontal asymptote y = 3 means that as x approaches positive or negative infinity, the function's output (y) approaches 3.
Now, let's evaluate each statement:
A) lim(x→3) f(x) = 0: This statement is not necessarily true since the function does not have an asymptote at x = 3. The asymptote is at y = 3, not x = 3.
B) lim(x→0) f(x) = 3: This statement is true. Since x ≠ 3, this asymptote does not affect the limit as x approaches 0. Therefore, the limit of the function as x approaches 0 is 3.
C) lim(x→∞) f(x) = 3: This statement is true. As x approaches positive or negative infinity, the function approaches y = 3.
D) lim(x→3) f(x) = ∞: This statement is not necessarily true since the asymptote at y = 3 does not mean that the function approaches infinity at x = 3.
Therefore, the correct answer is C. The statement that must be true when y = 3 is a horizontal asymptote of a rational function is:
C) lim(x→∞) f(x) = 3