Question

Evaluate the limit. $$ \lim _{t \rightarrow 0}\left(\frac{1}{t} \mathbf{i}+\cos t \mathbf{j}+\sin t \mathbf{k}\right) $$

Short answer

The limit of the given vector as \(t\) approaches 0 is undefined because the component in the 'i' direction is undefined. Other components evaluate to \(j + 0k\).

Step-by-step solution

Breakdown the Vector

The vector given has three parts: Its \(i\), \(j\) and \(k\) multiplies. Each can be treated as individual components and limits can be evaluated separately for them. Thus, the components are \(\frac{1}{t}\), \(\cos(t)\), \(\sin(t)\).

Evaluate Limit for 'i' Component

The 'i' component is \(\frac{1}{t}\). Let's try to find the limit as \(t\) approaches 0. As \(t\) approaches 0, \(\frac{1}{t}\) tends to infinity, therefore the limit for 'i' component is undefined.

Evaluate Limit for 'j' Component

The 'j' component is \(\cos(t)\). We know from basic trigonometry that the value of \(\cos(0)\) is 1. Therefore, the limit for 'j' component as \(t\) approaches 0 is 1.

Evaluate Limit for 'k' Component

The 'k' component is \(\sin(t)\). From basic trigonometry principles, we know that \(\sin(0) = 0\). Therefore, the limit for 'k' component as \(t\) approaches 0 is 0.