Question

for the function g whose graph is given, state the following. (if the answer is positive infinite, type \i\. if negative infinite, type
\. and if it does not exist, type \d\.) (a) lim x→∞ g(x) (b) lim x→−∞ g(x) (c) lim x→3 g(x) (d) lim x→0 g(x) (e) lim x→2 g(x) (f) the equations of the asymptotes (in increasing order). y = and y = x = and x = x =

Explanation:

Step1: Analyze $\lim_{x\to\infty}g(x)$

As $x$ approaches positive - infinity, the graph of $y = g(x)$ goes downwards. So, $\lim_{x\to\infty}g(x)=N$.

Step2: Analyze $\lim_{x\to-\infty}g(x)$

As $x$ approaches negative - infinity, the graph of $y = g(x)$ goes upwards. So, $\lim_{x\to-\infty}g(x)=I$.

Step3: Analyze $\lim_{x\to3}g(x)$

As $x$ approaches $3$ from both the left - hand and right - hand sides, the function values approach a finite value. So, $\lim_{x\to3}g(x)$ exists and is a real number. But since we are using the given symbols, and the function is well - behaved around $x = 3$, we assume it is not infinite. Let's say it's a non - infinite value. Here we just focus on the infinite/non - existent cases for the given symbols.

Step4: Analyze $\lim_{x\to2}g(x)$

As $x$ approaches $2$ from the left - hand side, $g(x)\to-\infty$ and as $x$ approaches $2$ from the right - hand side, $g(x)\to+\infty$. So, $\lim_{x\to2}g(x)$ does not exist, and we write $\lim_{x\to2}g(x)=D$.

Step5: Analyze $\lim_{x\to0}g(x)$

As $x$ approaches $0$ from both the left - hand and right - hand sides, the function values approach a finite value. So, $\lim_{x\to0}g(x)$ exists and is a real number. Let's assume it's not infinite for the purpose of using the given symbols.

Step6: Find vertical asymptotes

Vertical asymptotes occur where the function approaches infinity or negative infinity. From the graph, the vertical asymptotes are at $x=-2$, $x = 0$, and $x = 3$.

Answer:

(a) $N$
(b) $I$
(c) (a real - number value, not applicable in the given symbol set)
(d) $D$
(e) (a real - number value, not applicable in the given symbol set)
(f) $x=-2$, $x = 0$, $x = 3$