When Does a Limit Fail to Exist and Why is It Not at Positive Infinity?

Understanding the behavior of limits can be crucial in calculus, especially when dealing with functions that approach infinities. Often, it is observed that a limit can be considered non-existent and not at positive infinity, despite the graph suggesting the function grows without bound. This article explores these concepts in detail.

Why is the Limit Considered Non-Existent?

Both the statements 'the limit does not exist' and 'the limit is positive infinity' can be true simultaneously based on the appearance of the graph. However, these two statements have distinct meanings when examined through the lens of mathematical definitions and definitions from calculus textbooks.

Definition of a Limit

According to the official definition, for a limit limx→afxLlim_{x to a} fx L to be a real number L, the following must be true:

For any ε0varepsilon > 0, there exists a δ>0delta > 0 such that if 0|x-a|δ0 , then 0|fx-L|ε0 .

This means that as x→ax to a, fxfx gets arbitrarily close to LL.

Difference Between Limit Non-Existence and Infinity

When we write limx→afx∞lim_{x to a} fx infty or limx→afx-∞lim_{x to a} fx -infty, it means 'the limit does not exist ... but in a very specific way'. Infinity, whether positive or negative, is not a real number and it is not meaningful to state that fxfx is close to infinity. Instead, the definition is as follows:

For any real number MM, there exists a δ>0delta > 0 such that if 0|x-a|δ0 , then fxMfx > M.

Informally, this means we can make fxfx arbitrarily large by choosing input values sufficiently close to aa.

Consensus in Textbooks

Multiple calculus textbooks confirm that when we write limx→afx∞lim_{x to a} fx infty, we do not mean the limit exists but rather that it fails to exist in a particular way. This is quite consistent with the definition provided in Single Variable Calculus: One Variable Calculus, with Vector Spaces and One-Variable Calculus, with Vector Spaces by Holder, DeFranza, and Pasachoff, and further detailed in Single Variable Calculus: Concepts and Contexts by James Stewart where the authors state:

This does not mean that we are regarding ∞infty as a number. Nor does it mean that the limit exists. It simply expresses the particular way in which the limit does not exist.

Conclusion

While the graph may suggest the function tends towards positive infinity, this does not imply the limit exists. It suggests the limit is non-existent, but in a specific manner. It is critical in calculus to understand these subtle differences to accurately interpret and solve problems involving limits.