Manipulating Limits of Sequences

Calculus Level 4

Consider the following statements about positive functions $f(x)$ and $g(x)$ , whose limits to infinity exists:

A) $\lim_{ x \rightarrow \infty} f(x) = \lim_{ x \rightarrow \infty} g(x)$ .
B) $\lim_{ x \rightarrow \infty} f(x) - g(x) =0$ .
C) $\lim_{ x \rightarrow \infty} \sqrt{ f(x)} = \lim_{ x \rightarrow \infty} \sqrt{ g(x) }$ .

How many of the following 6 statements are true:

$A \Rightarrow B, B \Rightarrow C, C \Rightarrow A, A \Rightarrow C, B \Rightarrow A, C \Rightarrow B ?$

This is related to the discussion AM=GM when infinite?

5 4 6 3

1 solution

Calvin Lin Staff
Oct 15, 2014

(This is not a complete solution.)

If $\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} g(x) = L$ a finite value, then we can conclude that $\lim_{x \rightarrow \infty} f(x) - g(x) = 0$ .

However, if their limit is infinity, then we do not have any control over the limit of the difference. This is akin to saying that $\infty - \infty \neq 0$ .

As an explicit example:
To show that $A \not \Rightarrow B$ , take $f(x) = x, g(x) = 2x$ .
To show that $B \not \Leftarrow C$ , take $f(x) = x, g(x) = 2x$ .

I assumed existence of a limit meant that it had a finite value.

Jake Lai - 6 years, 6 months ago

This could be fixed by simply stating that $f$ and $g$ are real functions. In that case there would be no such thing as converging to infinity; that would be considered diverging.

Bart Nikkelen - 6 years, 8 months ago

The limit of a real function can be infinite.

I added the condition of a positive function so that $\sqrt{f(x) }$ will make sense.

There are some who do not consider converging to infinity as a valid property, and instead says that it "diverges to infinity".

Calvin Lin Staff - 6 years, 8 months ago

Manipulating Limits of Sequences

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