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Euge
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Find the limit $$\lim_{x\to \infty} x\left[\frac{1}{e} - \left(\frac{x}{x+1}\right)^x\right]$$
The limit of the function is infinity.
To find the limit at infinity, you can use the following steps:1. Simplify the function as much as possible.2. Identify the highest degree term in the numerator and denominator.3. Divide both the numerator and denominator by the highest degree term.4. Take the limit as x approaches infinity.
Finding the limit at infinity can help determine the long-term behavior of a function. It can also be used to find horizontal asymptotes, which can provide valuable information about the graph of the function.
Yes, the limit at infinity can be negative if the function approaches negative infinity as x approaches infinity.
Yes, there are other methods such as using L'Hopital's rule, using limits of sequences, or using substitution. However, these methods may not always work for every function and may require more advanced mathematical knowledge.