# How do you use the Squeeze Theorem to find #lim (arctan(x) )/ (x)# as x approaches infinity?

The limit is zero (see the reason using the Squeeze Theorem below).

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To use the Squeeze Theorem to find lim (arctan(x))/ (x) as x approaches infinity, we can establish two other functions that "squeeze" the given function and have known limits.

First, we can consider the function f(x) = 1/x. As x approaches infinity, f(x) approaches 0.

Next, we can consider the function g(x) = -1/x. As x approaches infinity, g(x) also approaches 0.

Now, we can compare the given function, lim (arctan(x))/ (x), with f(x) and g(x).

Since -1/x ≤ (arctan(x))/ (x) ≤ 1/x for all x > 0, and both f(x) and g(x) approach 0 as x approaches infinity, we can conclude that lim (arctan(x))/ (x) also approaches 0 as x 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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