How to compute limits at infinity?

In summary, the conversation discusses different methods for computing limits at infinity of functions. One method involves dividing both the numerator and denominator by the largest degree variable in the denominator, while another method involves looking at the highest powers in the numerator and denominator. The original problem is then solved, with one person getting the answer of negative infinity and another getting positive infinity. Graphing the function is suggested as a way to check the answer, and websites are recommended for graphing functions.
  • #1
mileena
129
0

Homework Statement



lim x→∞
##\frac{7x^2 + x + 11}{4 - x}##

Homework Equations


The Attempt at a Solution



I am sorry I am posting so much. But I think I have learned two different ways to compute limits at infinity of functions: one by the math lab tutor and another by the professor, but I am not sure. Sometimes they don't work.

The tutor said the denominator always controls. So always divide both the numerator and denominator by the largest degree variable in the denominator.

And my professor said that if the numerator has the highest power, the limit is ∞. If the denominator has the highest power, the limit is 0. And if the highest powers in the numerator and denominator equal, divide by the highest power.

So here is the problem:

lim x→∞
##\frac{7^2 + x + 11}{4 - x}##

It's too complicated for me now to learn to use LaTeX to write a fraction within a fraction, since I have to leave in 45 minutes and I am scrambling, but basically I divided each term above by x and not x2 (since the math lab tutor said the denominator always controls) and I got:

##\frac{∞ + 1 + 0}{0 - 1}## =

∞/-1 =

-∞

But the correct answer, according to my professor, is ∞.
 
Last edited:
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  • #2
I believe the correct answer is ##-\infty## unless you meant for x to go to negative infinity.
 
  • #3
DeadOriginal said:
I believe the correct answer is ##-\infty## unless you meant for x to go to negative infinity.

mileena : Don't be scared to post if you don't fully understand something, but really try to sit down and think about what's going on instead of rushing through and asking for help right away out of impatience.
 
Last edited:
  • #4
Something is horribly wrong because I'm getting large negative numbers.
 
  • #5
DeadOriginal said:
Something is horribly wrong because I'm getting large negative numbers.

Oh wow I read that as x-4 not 4-x aha. My bad.
 
  • #6
Hello!

If its a matter of checking if the answer is negative infinity or positive infinity, you
could always just try graphing it. And it does look like the answer is negative infinity.
 
  • #7
Zondrina said:
mileena : Don't be scared to post if you don't fully understand something, but really try to sit down and think about what's going on instead of rushing through and asking for help right away out of impatience.

Thanks for the tip. I am really high strung, so I get nervous very quickly.

I was going to ask if the sine function has a horizontal asymptote, but then I calmed down, looked it up, and found that the sine function doesn't begin to converge on one point, as you do with an asymptote, but it goes back and forth between 1 and -1. So, no, it doesn't have an asymptote.

As for the professor's answer in my original question, maybe I wrote it down wrong? I don't know.
 
  • #8
rkum99 said:
Hello!

If its a matter of checking if the answer is negative infinity or positive infinity, you
could always just try graphing it. And it does look like the answer is negative infinity.

Thank you rkum99! I had forgotten about using a graphing calculator. I just bought a TI-89, so it's pretty new to me.

Also, I learned there are at least two sites that will graph a function for you:

This one is pretty simple to use: http://www.fooplot.com
This one is a bit more challenging and more powerful : http://rechneronline.de/function-graphs/
 
  • #9
Use wolframalpha.com it has a load of other features than just graphing.
 
  • #10
Thanks Enigman! I will check that site out too.
 

1. What is a limit at infinity?

A limit at infinity is a mathematical concept that describes the behavior of a function as its input approaches infinity. It is used to determine the value that a function approaches as its input grows without bound.

2. How do you compute a limit at infinity?

To compute a limit at infinity, you must first simplify the function and then evaluate the limit as the input approaches infinity. This can be done by factoring, finding common denominators, or using algebraic manipulation.

3. What are the rules for computing limits at infinity?

There are several rules for computing limits at infinity, including the power rule, the quotient rule, and the sum and difference rules. These rules can be used to simplify the function and make it easier to evaluate the limit.

4. Can a limit at infinity be undefined?

Yes, a limit at infinity can be undefined if the function does not approach a finite number as its input approaches infinity. In this case, the limit is said to be infinite or does not exist.

5. Why is computing limits at infinity important?

Computing limits at infinity is important because it allows us to understand the behavior of functions as their input values grow without bound. This can help us make predictions and draw conclusions about the behavior of various phenomena in the real world.

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