# How do you use the Squeeze Theorem to find #lim tan(x)cos(sin(1/x))# as x approaches zero?

We need to consider right and left limits separately.

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To use the Squeeze Theorem to find the limit of tan(x)cos(sin(1/x)) as x approaches zero, we need to find two functions that "squeeze" the given function and have the same limit as x approaches zero.

First, we observe that -1 ≤ sin(1/x) ≤ 1 for all x ≠ 0. Therefore, -cos(1/x) ≤ cos(sin(1/x)) ≤ cos(1/x) for all x ≠ 0.

Next, we know that -1 ≤ cos(1/x) ≤ 1 for all x ≠ 0.

Since -1 ≤ -cos(1/x) ≤ cos(sin(1/x)) ≤ cos(1/x) ≤ 1 for all x ≠ 0, we can apply the Squeeze Theorem.

As x approaches zero, both -1 and 1 approach zero. Therefore, by the Squeeze Theorem, the limit of cos(sin(1/x)) as x approaches zero is also zero.

Finally, we can use the fact that the limit of tan(x) as x approaches zero is also zero. Therefore, the limit of tan(x)cos(sin(1/x)) as x approaches zero is zero.

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