Limits

[bt359]

\(\displaystyle \text{Evaluate: }\:\displaystyle{\lim_{x\to \infty}}\,\dfrac{4x^3\,-\,3x\,+\,1}{7x^3\,+\,2x^2\,-\,5}\)

Hello, This is from my tutorial sheet and the solution says the answer to above limit problem is 4/7, when we divide top and bottom by x^3. Because when we divide bottom and top with x^3 and then substitute x for infinity we get (4-0+0) / (7+0-0) = 4/7

However I was thinking the we did not need to divide top and bottom by x^3 because when we simply substitute for x we should get the answer to be -1/5. why cant i just subsisite x for infinity and get this: [4(0)-3(0)+1] / [7(0)+2(0)-5] = -1/5

Please can someone explain why i am wrong. Thank you in advance.

 

[Romsek]

\(\displaystyle \text{Evaluate: }\:\lim_{x\to1nfinity}\,\frac{4x^3\,-\,3x\,+\,1}{7x^3\,+\,2x^2\,-\,5}\)


Hello, This is from my tutorial sheet and the solution says the answer to above limit problem is 4/7, when we divide top and bottom by x^3. Because when we divide bottom and top with x^3 and then subsiture x for infinity we get 4-0+0 / 7+0-0 = 4/7

However I was thinking the we did not need to divide top and bottom by x^3 because when we simply substitute for x we should get the answer to be -1/5. why cant i just subsisite x for infinity and get this:
4(0)-3(0)+1 / 7(0)+2(0)-5 = -1/5


Please can someone explain why i am wrong. Thank you in advance.

In general working with 0 is a lot easier than working with infinity.

For example (0 - 0) = 0, (\(\displaystyle \infty\) - \(\displaystyle \infty\)) != 0, and \(\displaystyle \infty\)/\(\displaystyle \infty\) has no meaning.

Suppose you just had the limit of 4x³ / 7x³ and you plug infinity in for x.

You have 4*\(\displaystyle \infty\) / 7*\(\displaystyle \infty\) What is the value of this? It's not really defined. But if you divide both sides by x³ you get the easily interpreted 4/7.

As far as what you are doing how do you manage to get that substituting \(\displaystyle \infty\) for x in 4x³ for example gives you 4*0?

 

[HallsofIvy]

The problem is, of course, that you didn't "substitute infinity for x". You substituted 0 for x which is a very different thing.

The solution, as given, uses the fact that \(\displaystyle \lim_{x\to \infty} \frac{1}{x}= 0\) so that taking "x going to infinity" is the same as taking "1/x going to 0". But the change from x to 1/x is crucial.

 

[bt359]

The solution, as given, uses the fact that \(\displaystyle \lim_{x\to \infty} \frac{1}{x}= 0\) so that taking "x going to infinity" is the same as taking "1/x going to 0".

Ok I thought that when ever we see infinity, i just sub x for zero and I understand i was wrong. So will i be right to assume now that when ever i see x goes to infinity, I will assume \(\displaystyle \lim_{x\to \infty} \frac{1}{x}= 0\). That is I will try to make the expresion in terms of 1/x and then make 1/x=0. This should be correct right?

 

[Romsek]

Ok I thought that when ever we see infinity, i just sub x for zero and I understand i was wrong. So will i be right to assume now that when ever i see x goes to infinity, I will assume \(\displaystyle \lim_{x\to \infty} \frac{1}{x}= 0\). That is I will try to make the expresion in terms of 1/x and then make 1/x=0. This should be correct right?

yes. usually this means just dividing top and bottom by the largest power of x that appears. In this case you'd divide top and bottom by x³​.

 

[lookagain]

In general working with 0 is a lot easier than working with infinity.

For example (0 - 0) = 0,


(\(\displaystyle \infty\) - \(\displaystyle \infty\)) != 0, \(\displaystyle \ \ \ \)You have "!= 0" on the right. What do you mean?
I am asking for clarification, because I can't tell if you are stating that \(\displaystyle \ \ \ \infty - \infty = 0.\)*


and \(\displaystyle \infty\)/\(\displaystyle \infty\) has no meaning. \(\displaystyle \ \ \ \) **

* and ** \(\displaystyle \ \ \) Both \(\displaystyle \ \infty - \infty \ \ \ and \ \ \ \dfrac{\infty}{\infty} \ \ \) are indeterminate forms.

 

[Romsek]

* and ** \(\displaystyle \ \ \) Both \(\displaystyle \ \infty - \infty \ \ \ and \ \ \ \dfrac{\infty}{\infty} \ \ \) are indeterminate forms.

!= means not equal, and yes I pretty much stated those forms aren't defined.