Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 θcos θ

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Recognize that the limit involves a trigonometric function and a variable approaching zero. The expression given is \( \lim_{\theta \to 0} \theta \cos \theta \).

Understand that \( \cos \theta \) is a continuous function, and as \( \theta \to 0 \), \( \cos \theta \to \cos(0) = 1 \).

Rewrite the limit expression by separating the terms: \( \lim_{\theta \to 0} \theta \cos \theta = \lim_{\theta \to 0} \theta \cdot \lim_{\theta \to 0} \cos \theta \).

Substitute the known limit of \( \cos \theta \) as \( \theta \to 0 \), which is 1, into the expression: \( \lim_{\theta \to 0} \theta \cdot 1 \).

Evaluate the limit of \( \theta \) as \( \theta \to 0 \), which is 0. Therefore, the entire expression evaluates to \( 0 \cdot 1 = 0 \).

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