Question

Complete the table of values (to five decimal places), and use the table to estimate the value of the limit.

$\underset{x\to 0}{\lim }\frac{e^x-1}{x}$

Short answer

The value of limit of the given function is .

Step-by-step solution

Step 1. Given information.

The limit of function here given is,

$\underset{x\to 0}{\lim }\frac{e^x-1}{x}$

Step 2. Apply L’Hopital’s Rule.

Then,

$\begin{array}{c}\underset{x\to 0}{\lim }\left(\frac{e^x-1}{x}\right)\\ \underset{x\to 0}{\lim }\left(\frac{e^x}{1}\right)\\ \underset{x\to 0}{\lim }\left(e^x\right)\end{array}$

Step 3. Put the value.

Now, according to given table put the value of $x$in the function.

And get the value of$f\left(x\right)$

If$x=-0.1$, then$f\left(x\right)=0.99832$

If$x=-0.01$, then$f\left(x\right)=0.98312$

If$x=-0.001$, then$f\left(x\right)=0.97989$

If$x=0.001$, then$f\left(x\right)=0.98999$

If$x=0.01$, then$f\left(x\right)=0.99219$

If$x=0.1$, then$f\left(x\right)=0.99929$

Step 4. Complete the table.

Now, the completed table is below,

Therefore, on the basis of the values in the table we make the guess that,

$\underset{x\to 0}{\lim }\frac{e^x-1}{x}=1$