Question

Evaluate the limits in Exercises 42–45.

$\underset{t\to 1^{-}}{lim}(\ln t,\frac{e^t-1}{t-1},e^t)$

Short answer

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

Step-by-step solution

Step 1. Given Information.

The function:

$\underset{t\to 1^{-}}{lim}(\ln t,\frac{e^t-1}{t-1},e^t)$

Step 2. Apply the limits.

By applying the limit and solving,

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