Something weird about limit at infinity?

In summary, the problem involves finding the limit as x approaches infinity of f(x)/g(x), where f(x)=x^3 and g(x)=x^2. One solution from Spivak's Calculus shows that the limit does not exist, while the other solution simply states that the limit is infinity. However, since infinity is not a number, this still implies that the limit does not exist. The first solution may be overly complicated and the speaker is tired and appreciates any help.
  • #1
Elvz2593
5
0

Homework Statement


This is a problem I came up with when I was doing something similar in Spivak's Calculus; although a simpler version.


Suppose, we have [tex] f(x)=x^3 [/tex] and [tex]g(x)=x^2 [/tex]
find [tex]\lim_{x\rightarrow \infty} f(x)/g(x) [/tex]




Homework Equations


N/A


The Attempt at a Solution


So, I had 2 solutions for it. One says the limit exist, the other one says otherwise,

1. first method came from Spivak's book.

[tex]\lim_{x\rightarrow \infty} f(x)/g(x) = \lim_{x\rightarrow \infty} x^3/x^2 = \lim_{x\rightarrow \infty} (x^3/x^3)/(x^2/x^3) = \lim_{x\rightarrow \infty} 1/(1/x) [/tex]

let [tex] h(x)=1 [/tex] and [tex] j(x)=1/x [/tex]

[tex]\lim_{x\rightarrow \infty} f(x)/g(x) = \lim_{x\rightarrow \infty} h(x)/j(x) [/tex]
[tex]\lim_{x\rightarrow \infty} h(x)=1 [/tex]
[tex]\lim_{x\rightarrow \infty} j(x)=1/x=0 [/tex] this implies [tex] \lim_{x\rightarrow \infty} h(x)/j(x) [/tex] does not exist, otherwise [tex]\lim_{x\rightarrow \infty} h(x)= (\lim_{x\rightarrow \infty} h(x)/j(x) ) * ( \lim_{x\rightarrow \infty} j(x)) = 0 [/tex]
[tex]\lim_{x\rightarrow \infty} h(x) [/tex] can't both be 1 and 0.


2. the other method.

[tex] \lim_{x\rightarrow \infty} f(x)/g(x) = \lim_{x\rightarrow \infty} x^3/x^2 = \lim_{x\rightarrow \infty} x = \infty[/tex]



One of the solutions has to go. I probably just can't think straight since I have been up all night. Any help will be appreciated, my head is pretty much empty right now.:confused:
 
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  • #2
The first "method" is overly complicated but they are not contradictory. "[itex]\infty[/itex]" is not a number- saying that a limit is infinity is the same as saying the limit does not exist.
 
  • #3
HallsofIvy said:
The first "method" is overly complicated but they are not contradictory. "[itex]\infty[/itex]" is not a number- saying that a limit is infinity is the same as saying the limit does not exist.
Saved my day, I can go to bed now.:zzz:
 

Related to Something weird about limit at infinity?

1. What is the limit at infinity and how is it calculated?

The limit at infinity is a concept in calculus that refers to the value that a function approaches as its input approaches infinity. It is calculated by evaluating the function at larger and larger values, and observing the trend in the output values.

2. Can a function have a limit at infinity?

Yes, a function can have a limit at infinity if the output values approach a finite number as the input approaches infinity. This is known as a finite limit at infinity.

3. What is the difference between a finite limit at infinity and an infinite limit at infinity?

A finite limit at infinity means that the output values approach a finite number as the input approaches infinity. An infinite limit at infinity means that the output values either increase or decrease without bound as the input approaches infinity, and there is no finite limit.

4. How do you determine if a function has a finite or infinite limit at infinity?

To determine if a function has a finite or infinite limit at infinity, you can evaluate the function at increasingly large values of the input. If the output values approach a finite number, then the function has a finite limit at infinity. If the output values either increase or decrease without bound, then the function has an infinite limit at infinity.

5. Can a function have different limits at positive and negative infinity?

Yes, a function can have different limits at positive and negative infinity if the output values approach different numbers as the input approaches positive and negative infinity. In this case, the limit at infinity does not exist.

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