Evaluate: lim θ → π 4 cos ⁡ θ − sin ⁡ θ θ −...

Ayaan Barr

Ayaan Barr

Answered

2022-07-16

Evaluate: lim θ π 4 cos θ sin θ θ π 4

Answer & Explanation

Kroatujon3

Kroatujon3

Expert

2022-07-17Added 19 answers

Add and subtract cos ( π 4 ) = sin ( π 4 ) in the numerator to get:
lim θ π 4 cos θ sin θ θ π 4 = lim θ π 4 cos θ cos ( π 4 ) ( sin θ sin ( π 4 ) ) θ π 4
= lim θ π 4 cos θ cos ( π 4 ) θ π 4 lim θ π 4 sin θ sin ( π 4 ) θ π 4 = sin ( π 4 ) cos ( π 4 ) = 2
where we used the definition for derivatives of sine and cosine.
Gauge Terrell

Gauge Terrell

Expert

2022-07-18Added 5 answers

Let θ π 4 = x. Hence, θ = π 4 + x and
Hence,
lim θ π 4 cos θ sin θ θ π 4 = lim x 0 cos ( π 4 + x ) sin ( π 4 + x ) x = 2 lim x 0 sin x x = 2 .

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