Squeeze Theorem Exercises Directions: Determine the limit (if it exists). 1 𝑥

1. lim𝑥→0 𝑥 sin ( )

2. lim𝑥→0 𝑥 2 𝜋

1 𝑥

sin( )

3. Let 𝑓(𝑥) = 𝑥 if 𝑥 is an irrational number and 𝑓(𝑥) = 0 if 𝑥 is a rational number. Determine lim𝑥→0 𝑓(𝑥).

4. Let 𝑔(𝑥) = 1 + 𝑥 4 if 𝑥 is an irrational number and 𝑔(𝑥) = 1 + 2𝑥 2 if 𝑥 is a rational number. Determine lim𝑥→0 𝑔(𝑥). Hint: Assume |𝑥| ≤ 1.

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