Question

Represent pictorially that x log x is o($x^2$) by graphing x log x,$x^2$, and x$\log x/x^2$. Explain how this picture shows that x log x is o($x^2$).

Short answer

This question is based on concept of little o notation.

Step-by-step solution

Step 1:

Let us plot $x^2,x\log x\text{and}x\log x/x^2$

We see that$xlogx/x^2$ approaches to 0 when $x\ge 1$. Now to prove this we now use our little o notation.

Step 2:

Now, find the value$\underset{x\to \mathrm{\infty }}{lim} \frac{x\log x}{x^2}$.

$=\underset{x\to \mathrm{\infty }}{lim} \frac{\log x}{x}$

Since this is an$\frac{\infty }{\infty }$ form, we will use L’Hopital Rule.

Step 3:

Lastly, find the value of$\underset{x\to \mathrm{\infty }}{lim} \frac{\log x}{x}$.

$=\underset{x\to \mathrm{\infty }}{lim} \frac{1}{(\ln 10)}\ast \frac{1}{x}$

Since $\frac{1}{\mathrm{\infty }}\to 0$which can also be seen in the graph, therefore we can say that xlogx is o($x^3$)