Using a diagram of the unit circle and the Pythagorean Theorem, show that

Conclude that sin2 θ ≤ 2(1 − cos θ) ≤ θ2 and use this to give an alternative proof that the limit in Exercise 51 equals 0. Then give an alternative proof of the result in Exercise 54.

Data From Exercise 51

Explain why  involves an indeterminate form, and then prove that the limit equals 0.

Data From Exercise 54

Investigate  numerically or graphically. Then prove that the limit is equal to 1/2. See the proof of Theorem 2.

Transcribed Image Text:

sin² 0 < (1- cos 0)² + sin² 0 ≤ 0²