# How you slove this? #lim_(n->oo)cos(pisqrt(4n^2+n+1))#

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To solve the limit ( \lim_{n \to \infty} \cos(\pi \sqrt{4n^2 + n + 1}) ), we observe that as ( n ) approaches infinity, the expression inside the cosine function, ( \pi \sqrt{4n^2 + n + 1} ), also approaches infinity.

Since the argument of the cosine function approaches infinity, we know that the cosine function will oscillate between -1 and 1 infinitely as ( n ) increases. Therefore, the limit of ( \cos(\pi \sqrt{4n^2 + n + 1}) ) as ( n ) approaches infinity does not converge to a single value. Instead, it oscillates between -1 and 1 infinitely. Hence, the limit does not exist.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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