Question

Evaluate the limits in Exercises 42–45.

$\underset{t\to 0^{+}}{lim}(\frac{\sin t}{t},\frac{1-\cos t}{t},{(1+\frac{1}{t})}^t)$

Short answer

The evaluation of the limit $\underset{t\to 0^{+}}{lim}(\frac{\sin t}{t},\frac{1-\cos t}{t},{(1+\frac{1}{t})}^t)$is $(1,0,e)$.

Step-by-step solution

Step 1. Given Information.

The function:

$\underset{t\to 0^{+}}{lim}(\frac{\sin t}{t},\frac{1-\cos t}{t},{(1+\frac{1}{t})}^t)$

Step 2. Apply the limits.

By applying the limit and solving,

$$\begin{gathered} \underset{t\to 0^{+}}{lim}(\frac{\sin t}{t},\frac{1-\cos t}{t},{(1+\frac{1}{t})}^t)=\underset{t\to 0^{+}}{lim}\left(\frac{sint}{t}\right)i+\underset{t\to 0^{+}}{lim}\left(\frac{1-\cos t}{t}\right)j+\underset{t\to 0^{+}}{lim}{(1+\frac{1}{t})}^{t}k=1i+\sin 0j+ek \\ =i+ek \\ =(1,0,e) \end{gathered}$$