Evaluate the limit

by RoRi

August 25, 2015

Evaluate

$\lim_{t \to 0} (sin (2t) + t^2 \cos (5t)).$

We know (Example 3, p. 134 of Apostol) that sine and cosine are continuous. Further, we know that $t^2$ is continuous since it is a polynomial. Thus, by Theorem 3.2, $t^2 \cos (5t)$ is continuous since it is a product of continuous functions. By Theorem 3.2 again, $\sin (2t) + t^2 \cos (5t)$ is continuous since it is the sum of continuous functions. Therefore, we can compute the limit by evaluating at $t = 0$ ,