Question

Complete the table of values (to five decimal places), and use the table to estimate the value of the limit.$\underset{x\to 0}{\lim }\frac{\sin x}{x}$

Short answer

The value of limit of the given function is$1$ .

Step-by-step solution

Step 1. Given information.

The limit of function here given is,

$\underset{x\to 0}{\lim }\frac{\sin x}{x}$

Step 2. Apply L’Hopital’s Rule.

Then,

$\begin{array}{c}\underset{x\to 0}{\lim }\left(\frac{\sin \left(x\right)}{x}\right)\\ =\underset{x\to 0}{\lim }\left(\frac{\cos \left(x\right)}{1}\right)\\ =\underset{x\to 0}{\lim }\left(\cos \left(x\right)\right)\end{array}$

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$x=\pm 1$, then$f\left(x\right)=0.99984$

If$x=\pm 0.5$, then$f\left(x\right)=0.99996$

If $x=\pm 0.1$, then$f\left(x\right)=0.99909$

If $x=\pm 0.05$, then$f\left(x\right)=0.99996$

If $x=\pm 0.01$, then$f\left(x\right)=0.99999$

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,

$\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$