Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limt→0 2t / tan t

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Recognize that the limit involves a trigonometric function, specifically tan(t), which can be expressed in terms of sin(t) and cos(t). Recall that tan(t) = sin(t) / cos(t).

Rewrite the expression 2t / tan(t) as 2t / (sin(t) / cos(t)), which simplifies to 2t * (cos(t) / sin(t)).

This can be further simplified to (2t * cos(t)) / sin(t).

To apply the known limit lim(θ→0) sin(θ) / θ = 1, rewrite the expression as (2 * cos(t)) * (t / sin(t)).

Recognize that as t approaches 0, t / sin(t) approaches 1, and cos(t) approaches cos(0) = 1. Therefore, the limit can be evaluated by considering the product of these limits: 2 * 1 * 1.

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