Question

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

$\underset{x\to 0^{+}}{\lim }x\ln x$

Short answer

The value of limit of the given function is$0$ .

Step-by-step solution

Step 1. Given information.

The limit of function here given is,

$\underset{x\to 0^{+}}{\lim }x\ln x$

Step 2. Apply L’Hopital’s Rule.

Then,

$\begin{array}{c}\underset{x\to 0^{+}}{\lim }\left(x\ln \left(x\right)\right)\\ =\underset{x\to 0^{+}}{\lim }\left(\frac{\ln \left(x\right)}{\frac{1}{x}}\right)\\ =\underset{x\to 0^{+}}{\lim }\left(\frac{\frac{1}{x}}{-\frac{1}{x^2}}\right)\\ =\underset{x\to 0^{+}}{\lim }\frac{\frac{1}{x}}{-\frac{1}{x^2}}\\ \underset{x\to 0^{+}}{\lim }\left(-x\right)\end{array}$

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 $f\left(x\right)=-0.10000$

If $x=0.01$, then $f\left(x\right)=-0.01000$

If $x=0.001$, then $f\left(x\right)=-0.00100$

If $x=0.0001$, then $f\left(x\right)=-0.00010$

If $x=0.00001$, then $f\left(x\right)=-0.00001$

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,

$\underset{x\to 0^{+}}{\lim }x\ln x=0$