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Numerade Educator

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Problem 36 Easy Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \frac{e^{3x} - e^{-3x}}{e^{3x} + e^{-3x}} $

Answer

1

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Video Transcript

suppose you want to evaluate this limit. I know that if our function is a rational function involving E and our limit is that infinity the first thing you have to do is to factor out E was the highest exponents. And so from here we have limits as X approaches infinity of in this case between E raised to three X. And erase negative three X erase the three. Access the one with the highest exponents. So we have erase the three X. This times one minus, erase the negative six X. And then this all over the same process. We have erase the three X times one plus He raised the -6 X. From here we can reduce by getting rid of the common factor, erase the three X. And we have the limit as X approaches infinity of one minus erase negative six X Over one plus erase to -6 X. From here we Evaluate that infinity we have one erase to negative six times infinity over one plus erase to -6 times infinity And this is equal to 1 -1 erase the negative infinity Over one plus erase the negative infinity. And we know that erased the negative infinity approaches zero and so from here we have 1/1 which is equal to one. Therefore the value of the limit is one