# How do you use the Squeeze Theorem to find #lim Arctan(n^2)/sqrt(n)# as x approaches infinity?

See the explanation, below.

Consequently,

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To use the Squeeze Theorem to find the limit of Arctan(n^2)/sqrt(n) as n approaches infinity, we need to find two other functions that "squeeze" the given function and have the same limit as n approaches infinity.

First, we can observe that -1 < Arctan(n^2)/sqrt(n) < 1 for all positive values of n. This is because the range of the arctangent function is between -π/2 and π/2, and the denominator, sqrt(n), grows faster than the numerator, n^2, as n approaches infinity.

Next, we can find two functions, f(n) and g(n), such that f(n) ≤ Arctan(n^2)/sqrt(n) ≤ g(n) for all positive values of n. One possible choice is f(n) = -1 and g(n) = 1.

Now, we can take the limit of f(n), Arctan(n^2)/sqrt(n), and g(n) as n approaches infinity. Since f(n) and g(n) both have a limit of 1, by the Squeeze Theorem, the given function Arctan(n^2)/sqrt(n) also has a limit of 1 as n approaches infinity.

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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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