Question

I have a problem that says, as x goes toward infinity, the function approaches (?) The rational function is f(x)= 4x^2-1/4x^2-9
How do I figure out how to answer that!? Help please! =(

Answer 1

then you have juste to calculate 4x^2/4x^2 =1

Answer 2

Here's how to do it...
\[\Large \frac{4x^2-1}{4x^2 - 9}\]

Answer 3

can you tell me why? I don't really get it if i don't know why.

Answer 4

Figure out the largest exponent of x, whether in the numerator or the denominator.
..
so... what IS the largest exponent of x? ^_^

Answer 5

2

Answer 6

That's right. So what I want you to do now, is multiply that entire thingy with this *ugly* expression...
\[\Huge \frac{\frac1{x^2}}{\frac1{x^2}}\]

Answer 7

By the way, that *ugly* expression up there^
is effectively equal to 1

Answer 8

so just multiply it by 1?

Answer 9

Yeah... but a rather special incarnation of 1.
You'll see why later...

Answer 10

You can also do it by just ignoring everything but the highest powers of x on the numerator and denominator (these become the dominant terms as x gets larger)
in this case it's...
\[\Large \frac{ 4x^2 }{ 4x^2 }= \]

Answer 11

\[\huge \frac{4x^2 - 1}{4x^2 - 9}\cdot \frac{\frac1{x^2}}{\frac1{x^2}}\]
And this is just equal to
\[\Huge \frac{\frac{4x^2}{x^2}-\frac1{x^2}}{\frac{4x^2}{x^2}-\frac9{x^2}}\]
Which just looks horrible doesn't it? :P

Answer 12

i'm not very good at multiplying fractions and things. i'm pathetic really. failing college algebra for the 6th time.

Answer 13

But do you see what I did there? I just multiplied everything (in the numerator and denominator) by \(\Huge \frac1{x^2}\)

Answer 14

Then do my method @kawriter

Answer 15

Yeah, I suppose ^
I do love the magic, though :3

Answer 16

i'm not sure I see how you got what you got.

Answer 17

is that the only step @agent0smith ?

Answer 18

Look at the highest powers of x on the top and bottom. Ignore everything else.

Answer 19

You have to simplify what i gave.

Answer 20

okay

Answer 21

The thing is, @kawriter as x gets larger and larger, those 'lesser' exponents become more and more insignificant, and that the value of the expression depends more on the x with the highest exponent...so you really only have to check how that 'highest-exponent' term behaves ^_^

Answer 22

@agent0smith doI simplify it by square rooting the whole thing and coming out with 2/2 = 1?

Answer 23

crap! now i have to graph that function... If i'm asking too much of anyone, just let me know!

Answer 24

Yes.

Another example:

\[\Large y = \frac{ -2x^3 + 3x^2-x+1 }{ 4x^4-7x^3-5 }\]
First identify the highest power on the top and bottom.

Answer 25

3

Answer 26

and 4

Answer 27

Well what are the terms with the highest powers?

Graph: https://www.google.com/search?q=(4x%5E2-1)%2F(4x%5E2-9)&oq=(4x%5E2-1)%2F(4x%5E2-9)&aqs=chrome.0.69i57j69i62l3.265j0&sourceid=chrome&ie=UTF-8

Answer 28

\[\Large y = \frac{ -2x^3 + 3x^2-x+1 }{ 4x^4-7x^3-5 }\]

highest power of x on the top is the -2x^3. What about the bottom?

Answer 29

4^4

Answer 30

that graphing page is AMAZING!! THANK YOU!

Answer 31

You mean 4x^4?

Answer 32

yes sorry.

Answer 33

Now divide them like i did above, for the first question.

Answer 34

so would that be -1/2x?

Answer 35

Yep. And what is -1/2x close to if x is huge?

Answer 36

um, i have no idea.....3?

Answer 37

When you divide a small number by a huge number, what's the result close to?

Answer 38

the huge number

Answer 39

For example: https://www.google.com/search?q=1%2F1000000&oq=1%2F1000000&aqs=chrome.0.69i57j69i58j69i59j0.346j0&sourceid=chrome&ie=UTF-8

Answer 40

okay, i think.

Answer 41

So a small number divided by a huge number is close to...?

Answer 42

an even smaller number?

Answer 43

Yes, but you could round it to what?

Answer 44

1?

Answer 45

Well is a number like 0.000000001 closer to 0 or 1?

Answer 46

oh, 0

Answer 47

Which means -1/2x, if x is huge, is basically equal to...

Answer 48

0

Answer 49

Yep.

What about 10/2x^2, if x is huge, it must be basically...

Answer 50

0

Answer 51

so for my initial problem, the function approaches 0?

Answer 52

You worked out the answer to the original one earlier... "doI simplify it by square rooting the whole thing and coming out with 2/2 = 1?"

Answer 53

It's only equal to zero if AFTER simplifying, you end up with an x on the bottom, like -10/3x.

Answer 54

gotcha. i just did another one and I got 10/x so that equals 0?

Answer 55

Yes. And if you end up with something like x/5... what result do you get if you divide a huge number by a small number?

Answer 56

a really big number? cause x isn't on the bottom.

Answer 57

Yes, which just means the function approaches infinity as x approaches infinity.

Answer 58

coooooollllll!!! makes sense!

Answer 59

and the one i did earlier,

Answer 60

i had 4x^2/4x^2

Answer 61

so i took the square root of both, x^2's cancelled out and I was left with 2/2 which =1

Answer 62

You can't take the square root of anything, all you can do is cancel off common factors.

The 4s cancel, and the x^2 both cancel, leaving you with 1.

Answer 63

ohhhhhhhh.....gotcha. okay. why isn't it 0 if both cancel?

Answer 64

Well... is 4 divided by 4 equal to zero?

Answer 65

oh. you make sense sensei. lol. i'm super dumb in math! sorry!

Answer 66

Remember that x is just a number. So is x^2.

20x^5/20x^5 = 1

Answer 67

ahhh....cool.

Answer 68

thanks @agent0smith you are AWESOME!

Answer 69

You're welcome :)

Answer 70

can i ask you a new question?

Answer 71

Sure.

Answer 72

is the remainder theorem like synthetic division?

Answer 73

if it's not i'm so screwed. ha.

Answer 74

Well, you can use synthetic division to get a result/remainder, if that's what you mean.

Answer 75

okay, i think that's what i mean. heh. i'm given an equation and i have to use the theorem to find f(-3)

Answer 76

Yeah, you can use synth division. The remainder will be the value of f(-3)

Answer 77

oh okay cool.

Answer 78

starting synth division always messes me up. do you start out multiplying?

Answer 79

Like this: http://www.purplemath.com/modules/polysolv/remndr02.gif

The first coefficient, you just bring it down, then multiply by the number you're dividing by.

Answer 80

I got that here: http://www.purplemath.com/modules/remaindr.htm

Answer 81

would you mind checking it?

Answer 82

i'm so bookmarking that page! thanks

Answer 83

It has several examples of synthetic division at the bottom.

What was the actual question?

Answer 84

Given 9x^4 + 12x^3 +2x^2 -7x +74, find f(-3)

Answer 85

Did you get 518? http://www.wolframalpha.com/input/?i=+f%28x%29+%3D+9x%5E4+%2B+12x%5E3+%2B2x%5E2+-7x+%2B74%2C+find+f%28-3%29

Answer 86

no, I got 292 =(

Answer 87

Ahhh!!! I don't know what I did wrong! =(

Answer 88

You could take a pic of your work and upload it on photobucket or something, and paste the link here so i can check.

Answer 89

my phone can't take pictures. =(

Answer 90

Can I maybe describe my work?

Answer 91

i had the -15 as -39. wrong from the start. okay i'm going to work it one more time without looking at the picture.

Answer 92

Fixed an error