How to compute limits at infinity?

[mileena]

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 ∞.

 

[DeadOriginal]

I believe the correct answer is $-\infty$ unless you meant for x to go to negative infinity.

 

[STEMucator]

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.

 

[DeadOriginal]

Something is horribly wrong because I'm getting large negative numbers.

 

[STEMucator]

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.

 

[rkum99]

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.

 

[mileena]

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.

 

[mileena]

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/

 

[Enigman]

Use wolframalpha.com it has a load of other features than just graphing.

 

[mileena]

Thanks Enigman! I will check that site out too.