Question

Math question

Answer 1

Are you familiar with limits?
\[\lim_{t \rightarrow \infty} 2x^4-5x+1\]The issue here is that when we plug in \(\infty\) we end up with \(\infty\)-\(\infty\). Oops? So when we look at this, we should start by noticing the degree. The term with the highest degree means that it will reach that term faster than the other terms. In this case, \(2x^4\) is the term containing the highest degree. Therefore we can turn our limit into:
\[\lim_{t \rightarrow \infty}2x^4=\infty\]This method is how I was taught, but I believe there are other ways to logically think the problem through.

Answer 2

Yes, ok, so what do we do next?

Answer 3

So now do the same thing for \(t=-\infty\)

Answer 4

this page might help

http://www.purplemath.com/modules/polyends.htm

Answer 5

How did you get C? Can you show your work using limits so I can see your errors?

Answer 6

Well, doesn't have to be limits. I just want to see your logic is all

Answer 7

@CShrix I'm not really sure, I just assumed based on the advice you gave me. Can you explain?

Answer 8

Sure.

When we have a polynomial, the term with the highest degree determines the end behavior of its graph. Therefore, when we're given a polynomial and we want to know the end behavior, we can express it as a limit.
Let's take this for example:
\[\lim_{x \rightarrow \infty}x^4+x\]What this means, is that we want to see where y (aka f(x)) grows when x grows very very very very very large. Meaning, when we go infinitely to the right (\(x \rightarrow \infty\)) or to the left (\(x \rightarrow -\infty\)), how big is y?
In the case I have above, we want to know the end behavior when we go to the right. Do you understand thus far?

Answer 9

Yes, ok that makes sense.

Answer 10

So, it would be D?

Answer 11

Ok, great.
So since we now know that the term containing the highest degree (exponent) determines the end behavior of the graph, then that means that we don't care about the other terms, even the ones including other x's. This is fantastic, which means that we can express our limit as
\[\lim_{x \rightarrow \infty}x^4\]

Now let's look at the graph of \(x^4\): (See attachment)

What happens when x goes all the way to the right? We see the y continues to grow and even grows beyond what is shown. It grows, well, forever. Which means y grows to ∞ as x grows to ∞. Therefore, we can express this relationship in math terms using the limit. Therefore:
\[\lim_{x \rightarrow \infty}x^4=\infty\]

Answer 12

So now let's take your problem: \(2x^4-5x+1\)
We don't know what this graph looks like, and thankfully, we don't need to. We can apply our knowledge that the highest degree term will determine our end behavior. Therefore, we only need to think of the graph of \(Ax^4\)! Whether there is a coefficient, it doesn't really matter. So now we can model our limit as:
\[\lim_{x \rightarrow \infty}2x^4\]
You can do 2 things at this point:
1). Plug in \(\infty\) into the equation
2). In your head think of the graph of x^4 and see that as x goes to the right, y grows forever.
Therefore:
\[\lim_{x \rightarrow \infty}2x^4=\infty\]

Now let's look at
\[\lim_{x \rightarrow -\infty}2x^4\] (notice the negative underneath the limit). This means that we want to see what happens to y when we go LEFT in the graph. Well, once again, you can do this 2 different ways:
1). Plug in -infinity and realize that a negative to any even power is a positive. Therefore, we can infinity again.
2). Think of the graph of x^4 and realize that as you go to the left, y goes up forever.

Therefore
\[\lim_{x \rightarrow -\infty}2x^4=\infty\]

Answer 13

We have infinity again**

Answer 14

Sorry, the question wasn't loading for me. So the answer is B?

Answer 15

Yes, but I suggest that you go through the work again on your own to make sure you understand why.

Answer 16

Ok thanks for the help!