Question

given the following graph of a function f(x), determine (lim_{x \to infty} f(x)) and (lim_{x \to -infty} f(x)). (if an answer does not exist, enter dne.) (lim_{x \to infty} f(x) =) dne × (lim_{x \to -infty} f(x) =) dne ×
graph: y - axis with labels 10, 5, -5, -10; x - axis with labels -10, -5, 5, 10. a curve is plotted: on the right (x > 0) it rises steeply, on the left (x < 0) it falls steeply, crossing near the origin.
recall (lim_{x \to infty} f(x) = l) for (l in mathbb{r}) if and only if the function values get arbitrarily close to l as the values of x increase. otherwise, if the function valu...

Explanation:

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

As $x$ approaches positive infinity, observe the graph. The right - hand side of the graph (as $x$ gets very large positive) shows that the function values are increasing without bound (going to $+\infty$). In the context of limits, when a function's values grow without bound as $x$ approaches $\infty$, we say the limit is $+\infty$. But wait, the initial answer was DNE, but actually, for the limit as $x\to\infty$, if the function goes to $+\infty$ or $-\infty$, we can describe the limit in terms of infinity. Wait, no, let's re - examine the graph. Wait, the graph on the right (as $x$ increases) is going up, so as $x\to\infty$, $f(x)\to+\infty$. But maybe the problem considers that if it's going to infinity, does the limit exist? In some definitions, the limit as $x\to\infty$ of $f(x)$ is $+\infty$ (or $-\infty$) when the function values become arbitrarily large (positive or negative). But maybe in the context of this problem, let's check the left - hand side.

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

As $x$ approaches negative infinity, the left - hand side of the graph (as $x$ gets very large negative) shows that the function values are going down without bound (going to $-\infty$).

Wait, maybe I misread the initial graph. Wait, the graph: when $x$ is large positive (going to $\infty$), the function is increasing, so $\lim_{x\to\infty}f(x)=+\infty$? But the initial answer was DNE. Wait, maybe the problem's graph: let's look again. The graph on the right (x positive large) is a curve going up, so as $x\to\infty$, $f(x)$ approaches $+\infty$. On the left (x negative large), the curve is going down, so as $x\to-\infty$, $f(x)$ approaches $-\infty$. But in some cases, if we consider the limit in the real numbers, the limit does not exist (DNE) because $\infty$ is not a real number. Wait, the definition of a limit $\lim_{x\to\infty}f(x) = L$ where $L\in\mathbb{R}$. If $f(x)$ goes to $+\infty$ or $-\infty$, the limit does not exist (in the set of real numbers). But maybe the problem expects us to recognize the end - behavior. Wait, let's re - evaluate.

Wait, the graph: when $x$ approaches $+\infty$, the function is increasing, so it's going to $+\infty$. But since $+\infty$ is not a real number, the limit $\lim_{x\to\infty}f(x)$ does not exist (DNE)? No, wait, some textbooks define the limit as $+\infty$ when the function grows without bound. But the initial answer was marked wrong. Wait, maybe I made a mistake. Let's check the graph again. The right - hand part of the graph (x positive) is a curve that is increasing, so as $x$ gets larger and larger (approaches $\infty$), $f(x)$ gets larger and larger, so $\lim_{x\to\infty}f(x)=+\infty$. But if the problem considers that the limit must be a real number, then it DNE. For the left - hand side, as $x$ approaches $-\infty$, the function is decreasing, going to $-\infty$, so $\lim_{x\to-\infty}f(x)=-\infty$, which also DNE in the real - number limit sense. But maybe the problem's graph is different. Wait, the user's graph: the right side (x positive) is a curve going up, left side (x negative) is a curve going down. So:

For $\lim_{x\to\infty}f(x)$: As $x$ increases without bound, $f(x)$ increases without bound, so the limit is $+\infty$ (but if we consider the limit in $\mathbb{R}$, it DNE). For $\lim_{x\to-\infty}f(x)$: As $x$ decreases without bound, $f(x)$ decreases without bound, so the limit is $-\infty$ (or DNE in $\mathbb{R}$). But the initial answers were marked wrong. Wait, maybe I misread the graph. Wait, maybe the right - hand side is not going to $\infty$? No, the graph shows that as $x$ moves to the right (positive x - direction), the y - value is going up, so it's increasing without bound. As $x$ moves to the left (negative x - direction), the y - value is going down, decreasing without bound.

But according to the limit definition, if $\lim_{x\to\infty}f(x)=L$ where $L\in\mathbb{R}$, then $L$ must be a real number. Since $f(x)$ goes to $+\infty$ (not a real number), $\lim_{x\to\infty}f(x)$ DNE. Similarly, $\lim_{x\to-\infty}f(x)$ DNE because it goes to $-\infty$ (not a real number). But the initial answers were marked wrong, so maybe there's a misinterpretation. Wait, maybe the graph has a horizontal asymptote? No, the graph on the right is increasing, left is decreasing.

Wait, perhaps the problem is that when $x\to\infty$, the function is going to $+\infty$, so we write $\lim_{x\to\infty}f(x)=+\infty$ (and similarly $\lim_{x\to-\infty}f(x)=-\infty$), but in some contexts, these are considered as the limits (even though they are not real numbers). Let's check the recall statement: "Recall $\lim_{x\to\infty}f(x)=L$ for $L\in\mathbb{R}$ if and only if the function values get arbitrarily close to $L$ as the values of $x$ increase. Otherwise, if the function values..." So if the function values don't get close to a real number $L$, the limit does not exist. So in that case, both limits DNE? But the initial answers were marked wrong, so maybe I made a mistake. Wait, maybe the graph is different. Wait, the user's graph: let's see, the right - hand side (x positive) is a curve that is increasing, so as $x\to\infty$, $f(x)\to+\infty$. The left - hand side (x negative) is a curve that is decreasing, so as $x\to-\infty$, $f(x)\to-\infty$. So according to the recall, since $+\infty$ and $-\infty$ are not real numbers, the limits do not exist. So the answers should be DNE for both? But the initial answers were marked wrong. Wait, maybe the graph is actually showing that as $x\to\infty$, the function approaches a horizontal asymptote? No, the graph is going up. Maybe the user made a mistake in the graph, but based on the given graph, let's proceed.

Answer:

$\lim_{x \to \infty} f(x)=\infty$ (or DNE, but if we consider the extended real number system, it's $+\infty$) and $\lim_{x \to -\infty} f(x)=-\infty$ (or DNE). But according to the recall, since $L$ must be in $\mathbb{R}$, both limits DNE. So:

$\lim_{x \to \infty} f(x)=\boxed{\infty}$ (or DNE, but if we follow the extended real numbers) and $\lim_{x \to -\infty} f(x)=\boxed{-\infty}$ (or DNE). Wait, but the initial answer was DNE. Maybe the correct answers are $\lim_{x\to\infty}f(x)=\infty$ and $\lim_{x\to-\infty}f(x)=-\infty$. Let's re - check the limit definition. In some calculus courses, the limit as $x\to\infty$ of $f(x)$ is $+\infty$ if for every positive number $M$, there exists a number $N$ such that if $x > N$, then $f(x)>M$. Similarly, for $-\infty$. So in that case, $\lim_{x\to\infty}f(x)=+\infty$ and $\lim_{x\to-\infty}f(x)=-\infty$.

So correcting the initial answers:

$\lim_{x \to \infty} f(x)=\boxed{\infty}$ (or $+\infty$)

$\lim_{x \to -\infty} f(x)=\boxed{-\infty}$