Make conjectures about the following limits using the given graphs.
\lim\limits_{x \to\infty }f\left(x\right)
We want to make a conjecture about \lim\limits_{x \to \infty} f\left(x\right). That means observing what the function does as x gets larger and larger.
From what we see on the graph, as x gets bigger, the function values get closer to 4.
From the graph, we can conjecture:
\lim\limits_{x \to\infty }f\left(x\right)=4
\lim\limits_{x \to \infty} f\left(x\right)
We see in the graph that as x gets larger and larger, the graph of f\left(x\right) gets closer and closer to 0, from both the positive and negative side.
The oscillations do not matter, because the value that f\left(x\right) seems to be tending towards is 0.
From the graph, we can conjecture:
\lim\limits_{x \to \infty} f\left(x\right) = 0
\lim\limits_{x \to \infty} f\left(x\right)
We see in the graph that as x gets larger and larger, the graph of f\left(x\right) also keeps increasing in value. Because the function keeps increasing, the conjecture is that:
\lim\limits_{ x \to \infty} f\left(x\right) = +\infty
We cannot choose that this limit is a positive number, since \infty isn't a number itself.
From the graph, we can conjecture:
\lim\limits_{x \to \infty} f\left(x\right) = +\infty
\lim\limits_{x \to \infty} f\left(x\right)
Since we are interested in the \lim\limits_{x \to \infty} f\left(x\right), we want to know what happens to the function as x gets larger and larger.
In this case, x is small when:
f\left(x\right) = 4
As x gets bigger, on the graph we see that f\left(x\right) has the value of 2, and doesn't appear to change.
From the graph, we can conjecture:
\lim\limits_{x \to \infty} f\left(x\right) = 2\\
\lim\limits_{x \to \infty} f\left(x\right)
Since we are interested in the \lim\limits_{x \to \infty} f\left(x\right), we want to know what happens to the function as x the input value, gets larger and larger.
Notice from the graph that as the input or x values increase, the function value decreases, getting more and more negative. If this trend continues, the function will keep reaching more negative values.
From the graph, we can conjecture:
\lim\limits_{x \to \infty} f\left(x\right) = -\infty\\\\
\lim\limits_{x \to \infty} f\left(x\right)
In the graph we see that f\left(x\right) appears to be an increasing function, but never surpasses the value of 15, even as the input, or x values, get bigger and bigger.
It also does not appear to get farther away from 15 after getting close to it.
From the graph, we can conjecture:
\lim\limits_{x \to \infty} f\left(x\right) = 15
\lim\limits_{x \to 5} f\left(x\right)
In the graph we see that f\left(x\right) oscillates for all values of x. However, as the input values of f\left(x\right) get closer to 5 from either side, we see that f\left(x\right) gets closer to 0.
From the graph, we can conjecture:
\lim\limits_{x \to 5} f\left(x\right) = 0
\lim\limits_{x \to \infty} f\left(x\right)
In the graph we see that f\left(x\right) oscillates for all values of x.
This continues as x gets larger and larger, so that f\left(x\right) never seems to get close to one particular value. It just continues to oscillate as before. If this pattern continues, our best conjecture that the limits of the function does not exist.
From the graph, we can conjecture that \lim\limits_{x \to \infty} f\left(x\right) does not exist.