Question

for x ≥ 0, the horizontal line y = 2 is an asymptote for the graph of the function f. which of the following statements must be true?
a f(0)=2
b f(x)=2 for all x ≥ 0
c f(2) is undefined.
d $lim_{x
ightarrow2}f(x)=infty$
e $lim_{x
ightarrowinfty}f(x)=2$

Explanation:

Step1: Recall definition of horizontal asymptote

A horizontal asymptote $y = L$ for $x\geq0$ of the function $y = f(x)$ means $\lim_{x
ightarrow\infty}f(x)=L$ when $x\geq0$. Here $L = 2$.

Step2: Analyze each option

• Option A: Just because $y = 2$ is an asymptote, we cannot say $f(0)=2$. The function value at $x = 0$ has no relation to the asymptote value.
• Option B: $f(x)$ is not necessarily equal to 2 for all $x\geq0$. An asymptote is a value that the function approaches as $x$ goes to infinity, not a constant - value of the function.
• Option C: There is no reason to conclude that $f(2)$ is undefined. The asymptote $y = 2$ has no direct connection to the value of the function at $x = 2$.
• Option D: $\lim_{x

ightarrow2}f(x)=\infty$ is wrong. The asymptote information has nothing to do with the limit as $x$ approaches 2.

• Option E: Since $y = 2$ is a horizontal asymptote for $x\geq0$ of the function $f(x)$, by the definition of a horizontal asymptote, $\lim_{x

ightarrow\infty}f(x)=2$.

Answer:

E. $\lim_{x
ightarrow\infty}f(x)=2$