Find the limit. lim_{t to 0} ( {e^{-3t} mathbf{i} + frac{t^2}{sin^2 t} mathbf{j} + sin 3t ...


Find the limit.

{eq}\lim_{t \to 0} \left ( {e^{-3t}\; \mathbf{i} + \frac{t^2}{\sin^2 t} \;\mathbf{j} + \sin 3t \; \mathbf{k}} \right ) {/eq}

Answer and Explanation:

{eq}\text{Let's find the limit of each component one by one:}\\ \lim_{t\rightarrow 0}e^{-3t}\\ \text{Applying the limits:}\\ =1\\ \lim_{t\rightarrow 0}\frac{t^2}{\sin^2t}\\ \lim_{t\rightarrow 0}\frac{1}{\left (\frac{\sin t}{t} \right )^2}\\ \text{Applying the standard limit:}\\ \lim_{x\rightarrow 0}\frac{\sin x}{x}=1\\ \text{Applying we get:}\\ =1\\ \lim_{t\rightarrow 0}\sin 3t\\ \text{Applying the limits:}\\ =0\\ \text{so the final answer is:}\\ <1,1,0> {/eq}

Learn more about this topic:

Understanding the Properties of Limits

from Math 104: Calculus

Chapter 7 / Lesson 5

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