Question

the function g is continuous at all x except x = 2. if lim_{x→2}g(x)=∞, which of the following statements about g must be true?
a g(2)=∞
b the line x = 2 is a horizontal asymptote to the graph of g.
c the line x = 2 is a vertical asymptote to the graph of g.
d the line y = 2 is a vertical asymptote to the graph of g.

Explanation:

Step1: Recall limit and asymptote definitions

The limit $\lim_{x
ightarrow a}f(x)=\infty$ means that as $x$ approaches the value $a$, the function values $f(x)$ grow without bound. A vertical asymptote of a function $y = f(x)$ occurs at $x = a$ when $\lim_{x
ightarrow a^-}f(x)=\pm\infty$ or $\lim_{x
ightarrow a^+}f(x)=\pm\infty$ or $\lim_{x
ightarrow a}f(x)=\pm\infty$. Here $a = 2$ and $\lim_{x
ightarrow 2}g(x)=\infty$.

Step2: Analyze each option

• Option A: The value of a function at a point where the limit is infinity is not defined as infinity. A function value $g(2)$ is a real - valued output for a well - defined function at $x = 2$, and $\infty$ is not a real number. So $g(2)=\infty$ is not correct.
• Option B: A horizontal asymptote is of the form $y = c$ (where $c$ is a constant) and is related to the behavior of the function as $x

ightarrow\pm\infty$, not as $x$ approaches a finite value. So $x = 2$ cannot be a horizontal asymptote.

• Option C: Since $\lim_{x

ightarrow 2}g(x)=\infty$, by the definition of a vertical asymptote, the line $x = 2$ is a vertical asymptote to the graph of $g$.

• Option D: A vertical asymptote is of the form $x = k$ (where $k$ is a constant), not $y=k$. So $y = 2$ cannot be a vertical asymptote.

Answer:

C. The line $x = 2$ is a vertical asymptote to the graph of $g$.