What is $$\lim_{x\to \infty} \left (1+\frac{1}{x}\right )^{-x}?$$

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BlackTom AI · Accepted Answer

The prompt presents a limit question and indicates answer choices, but the provided data shows that the list of answer options is missing (the answer_options array is empty), while the answer field contains a single option: a. 1/e.

First, restating the question: What is lim_{x -> ∞} (1 + 1/x)^{-x} ? This asks for the value of a classic indeterminate-looking expression as x grows without bound.

Now, evaluating the mathematical idea behind the limit: a well-known related limit is lim_{x -> ∞} (1 + 1/x)^x = e. The given expression flips the exponent sign, so (1 + 1/x)^{-x} = 1 / (1 + 1/x)^x. Therefore, as x becomes very large, (1 + 1/x)^x approaches e, and its reciprocal approaches 1/e. This reasoning relies on the standard exponential limit that defines e, and it matches the transformation of an exponent from positive to negative.

Since the answer_options list is empty, we cannot perform a choice-by-choice analysis across multiple alternatives. In a typical multiple-choice setup, each option would be weighed against this standard limit result, checking for equivalence to 1/e, or for plausible misinterpretations (such as confusing the sign in the exponent or misapplying the limit of the base). For example, some common misconceptions might include:
- Thinking the limit yields e instead of 1/e due to not accounting for the negative exponent, which would be incorrect because raising to a negative exponent corresponds to taking a reciprocal.
- Assuming the limit diverges or approaches 0 due to intuition about small increments, which would be incorrect given the precise relationship to the standard e-limit.

Given the standard result, option a. 1/e is the value that aligns with the mathematical limit. However, without the other choices to compare, we cannot discuss why those specific alternatives would be right or wrong.

If you can provide the missing answer options, I can expand this into a complete, option-by-option analysis, citing exact reasoning for each choice and highlighting common pitfalls for incorrect ones.

MTH1030 -1035 - S1 2025 MTH1030/35 Week 7 lesson quiz: L’Hôpital’s rule

What is \[\lim_{x\to \infty} \left (1+\frac{1}{x}\right )^{-x}?\]

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