Something weird about limit at infinity?

[Elvz2593]

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 $$f(x)=x^3$$ and $$g(x)=x^2$$
find $$\lim_{x\rightarrow \infty} f(x)/g(x)$$

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.

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

let $$h(x)=1$$ and $$j(x)=1/x$$

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


2. the other method.

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



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.

 

[HallsofIvy]

The first "method" is overly complicated but they are not contradictory. "$\infty$" is not a number- saying that a limit is infinity is the same as saying the limit does not exist.

 

[Elvz2593]

The first "method" is overly complicated but they are not contradictory. "$\infty$" 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: