Is (1^{-infty}) Indeterminate?

The expression 1-infty is often discussed in the realm of limits and sequences. While similar forms such as 0/0 and (infty/infty) are considered indeterminate, the expression 1-infty is not. This article delves into why 1-infty is considered an indeterminate form and explores the implications of this concept.

Understanding (1^{-infty})

The expression 1x for any real number (x), including negative infinity, always equals 1. Therefore, it stands that: [1^{-infty} 1.]

However, when dealing with limits, the situation can become more complex. For example, if we have the expression (a_n^{b_n}) where (lim_{n to infty} a_n 1) and (lim_{n to infty} b_n -infty), then the limit [lim_{n to infty} a_n^{b_n}] is considered indeterminate. This means that without further analysis, it is impossible to determine the exact value of this limit just from the given information.

Indeterminate Forms in Calculus

There are several well-known indeterminate forms in calculus, including:

(frac{0}{0}) (frac{infty}{infty}) (infty times 0) (1^infty) (0^0) (infty^0) (infty - infty)

Specifically, 1^infty and 1^{-infty} are both considered indeterminate forms. This is because the limit of the form ab can vary based on the specific nature of the functions involved.

For instance, consider the following limits:

[lim_{n to infty} left(1 - frac{1}{n}right)^{-n} e^{-1}] [lim_{n to infty} left(1 frac{1}{n}right)^{-n} e] [lim_{n to infty} left(1 - frac{1}{n}right)^{-n^2} infty] [lim_{n to infty} left(1 frac{1}{n^2}right)^{-n} 1.]

In each case, the expression (a_n^{b_n}) is in the form of (1^infty), but the resulting limits are quite different.

Sequences and Indeterminate Forms

When dealing with sequences, the indeterminate form (1^{-infty}) can be tricky. Suppose we have sequences (a_n 1) and (b_n to -infty). The limit [lim_{n to infty} 1^{b_n} 1^{-infty}] is indeterminate because, without further information, we cannot determine the exact value of the limit.

For example, consider the limit:

[lim_{x to 0} 1x^{-1/x} frac{1}{e}.]

This shows that the value of (1^{-infty}) is not universally 1, but rather depends on the specific forms of the sequences involved. Therefore, (1^{-infty}) is indeed an indeterminate form.

Conclusion

While the expression (1^{-infty}) is not indeterminate in the same sense as other forms like (frac{0}{0}), it is an indeterminate form when considered within the context of limits of functions. This indeterminacy highlights the importance of careful analysis when dealing with limits involving the form (1^x) as (x) approaches (-infty).