How to find the limit of functions at $x=0$ when drawing graphs?

Question

For example, sketch $y=x(\ln x)^2$.

What is not immediately obvious is the limit of $y$ as $x$ approaches $0$ from the right.

The same goes for $y=x^x$.

Is there a way, in general, to find out the behavior of such curves as $x$ approaches zero?

Answer 1

Hints:

Re-write $x$ln$(x)^2$ as $\frac{x}{\frac{1}{lnx^2}}$ and apply L'Hopital's Rule.

Re-write $x^x$ as $e^{xln(x)}$ and solve.

Alternatively, you could do as you said (graph y= the expressions with a graphing calculator), and find the y value as x approaches 0 for both problems.

Answer 2

Assuming you know the limit $$\lim_{t\to+\infty}\frac{\ln(t)}{t}=0$$ rerwrite $x(\ln(x))^2$ as $(\sqrt{x}\ln(x))^2$ and then $\left(-2\frac{\ln\left(\frac{1}{\sqrt{x}}\right)}{\frac{1}{\sqrt{x}}}\right)^2$. Now take $t=\frac{1}{\sqrt{x}}$. Note that the function $x\to x^2$ is continious on $\mathbb{R}$ and since $x\to 0^+$ so we have $t\to+\infty$.