Do Functions Tending to Infinity Always Result in Infinity When Combined?

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Discussion Overview

The discussion revolves around the behavior of functions that tend to infinity when combined through various operations such as addition, multiplication, subtraction, and division. Participants explore the implications of these operations in the context of limits and provide examples to illustrate their points.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that if both functions f(x) and g(x) tend to infinity, then f(x) + g(x) should also tend to infinity, but seeks proof for this assertion.
  • Another participant provides a reasoning approach, stating that if g(x) tends to infinity, then for sufficiently large x, g(x) exceeds any positive number N, leading to the conclusion that f(x) + g(x) tends to infinity if f(x) also tends to infinity.
  • A third participant clarifies the distinction between limits at infinity and infinite limits, reiterating that if both f(x) and g(x) approach infinity, then f(x) + g(x) should also approach infinity, and provides a proof structure for multiplication as well.
  • Concerns are raised about the operations f(x) - g(x) and f(x)/g(x), with a participant noting that these do not necessarily yield defined results, as infinity minus infinity and infinity over infinity are undefined forms.
  • One participant requests further clarification on the implications of a previous comment regarding the behavior of f - g and f/g when both functions are equal and tend to infinity.
  • Another participant reiterates that if f = g, then both f - g and f/g yield 0 and 1, respectively, affirming the earlier point made.

Areas of Agreement / Disagreement

Participants express differing views on the outcomes of certain operations involving functions that tend to infinity, particularly regarding subtraction and division. There is no consensus on the implications of these operations, and the discussion remains unresolved.

Contextual Notes

Participants highlight the importance of definitions and the conditions under which the limits are considered, indicating that the discussion may depend on specific interpretations of limits and the behavior of functions near infinity.

Juggler123
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Suppose that the two functions f(x) and g(x) both tend to infinity then surely f(x) + g(x) also tends to infinity? How can you prove this though? Similarly f(x)*g(x) would also tend to infinity wouldn't it? f(x) - g(x) and f(x)/g(x) wouldn't tend to anything though surely since infinity minus infinity and infinity over infinity are both undefined. Can anyone help me?
 
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To help you a bit with the first one.

Saying that g(x) tends to infinity certainly implies the less strict interpretation, namely that for x>X, we have g(x)>N, where X and N are both some positive numbers.

Thus, for all x>X, we have the inequality

f(x)+g(x)>f(x)+N>f(x)

Thus, if f(x) is tending to infinity, then surely, f+g also.
 
"limit of functions at infinity" implies you are talking about the limit as x goes to infinity but you are actually asking about infinite limits.
Juggler123 said:
Suppose that the two functions f(x) and g(x) both tend to infinity then surely f(x) + g(x) also tends to infinity? How can you prove this though?
Suppose [itex]\lim_{x\to a} f(x)= \infty[/itex] and [itex]\lim_{x\to a} g(x)= \infty[/itex]. Then, given any Y> 0, there exist [itex]\delta_1> 0[/itex] such that if [itex]|x-a|< \delta_1[/itex] then f(x)> Y and there exist [itex]\delta_2< 0[/itex] such that if [itex]|x- a|< \delta_2[/itex] then g(x)> 0. Take [itex]\delta[/itex] to be the smaller of [itex]\delta_1[/itex] and [itex]\delta_2[/itex] so that if [itex]|x-a|< \delta[/itex] both [itex]|x-a|< \delta_1[/itex] and [itex]|x-a|< \delta_2[/itex] are true. Then f(x)> Y and g(x)> 0 so f(x)+ g(x)> Y+ 0.

Similarly f(x)*g(x) would also tend to infinity wouldn't it?
Similar proof. Given any real number, Y, there exist real number [itex]\delta[/itex] such that if [itex]|x-a|< \delta[/itex], [itex]f(x)> Y[/itex] and [itex]g(x)> 1[/itex]. Then f(x)g(x)> (Y)(1).

f(x) - g(x) and f(x)/g(x) wouldn't tend to anything though surely since infinity minus infinity and infinity over infinity are both undefined. Can anyone help me?
Let f(x)= g(x)= 1/(x-a). Then f and g both tend to infinity as x goes to a but f- g and f/g tend to 0 and 1 respectively.
 
Could someone delineate/prove HallsofIvy's last comment regarding f-g and f/g in more detail?
 
f=g so f-g=0 and f/g=1...
 
woodyallen1 said:
f=g so f-g=0 and f/g=1...
Excellent dilineation!
 
Very Kind of you PF..
 

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