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Problem 32 Easy Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} (e^{-x} + 2 \cos 3x) $

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Video Transcript

to evaluate the limits of erase the negative X. Plus to co sign of three X. As X approaches infinity, we first write this into the limit as X approaches infinity of erase the negative X. Plus you have a limit as X approaches infinity Of to co sign of three x. Now evaluating the first limit we have erased negative infinity Plus we have rewriting the second limit. That's too the limit. As X approaches infinity of Co sign of three x. Note that co sign function is a periodic function and we know that the value of co sign range is from negative 1 to 1. That means co sign of X. This is greater than or equal to negative one But this is less or equal to one. And because this is periodic The Value of Co Sign of three X. As X gets larger and larger Will either be -1 or one. and because the value of coastline of three x approaches to two values -1 or one. Then the value of the whole limit is indeterminate. That means it does not exist. In order to have a limit, we only need one value