Answer:
the limit of sin(1/θ) as θ approaches 0 is equal to 0.
Explanation:
To evaluate this limit:
lim θ→0 sin(1/θ)
we can use the squeeze theorem. First, we note that -1 ≤ sin(1/θ) ≤ 1 for all values of θ, since the sine function oscillates between -1 and 1.
Next, we consider the limit of two other functions, -1/|θ| and 1/|θ|, as θ approaches 0:
lim θ→0 -1/|θ| = -∞
lim θ→0 1/|θ| = ∞
Since sin(1/θ) is always between -1/|θ| and 1/|θ|, we can apply the squeeze theorem to conclude that:
lim θ→0 sin(1/θ) = 0
Therefore, the limit of sin(1/θ) as θ approaches 0 is equal to 0.