Does Arctanh Go to Infinity When Its Argument Approaches Infinity?

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SUMMARY

The discussion centers on the behavior of the arctanh function as its argument approaches infinity. It is established that for real numbers, the hyperbolic tangent function (tanh) is bounded by -1 and 1, meaning that if the argument of arctanh exceeds this range, the output becomes complex. Therefore, arctanh does not go to infinity for real arguments but transitions to complex values when the input is outside the domain of [-1, 1].

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  • Understanding of hyperbolic functions, specifically tanh and arctanh.
  • Knowledge of complex numbers and their properties.
  • Familiarity with limits in calculus.
  • Basic understanding of function domains and ranges.
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rmiller70015
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So, I'm doing a problem where I take arctanh to a limit, and I was wondering if the arctanh function goes to infinity if the argument inside of the function goes to infinity when passing through the limit.
 
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Is the argument real or complex? For real numbers, |tanh| ≤ 1. If the argument for arctanh is not in the domain [-1,1], then arctanh must be complex.
 

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