51: When x approaches infinity, then the first two terms together yield approximately 13, to this we then add a term that oscillates between 1 and −1, so the whole expression oscillates and the amplitude does not decrease to zero. We intuitively conclude that the given limit does not exist. One way to prove this is to use Heine's theorem.

Prepare two sequences of values of x going to infinity that will capture two extremes of behaviour of the given expression.

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