The diagrams below show the graphs of $x^2-4x^{\frac{1}{2}}$, $x^2$ and $4x^{\frac{1}{2}}$; on the left-hand side for the interval $[0,8]$ and on the right-hand side for the interval $[40,48]$.

By comparing the graphs we see that the graph of $f(x)=x^2-4x^{\frac{1}{2}}$ for $x$ near $0$ resembles the graph of $-4x^{\frac{1}{2}}$ and looks more like the graph of $x^2$ for large values of $x$ .

In other words, the $f(x)=x^2-4x^{\frac{1}{2}}$ function resembles $x\mapsto -4x^{\frac{1}{2}}$ for $x$ near $0$ and is more similar to the function $x\mapsto x^2$ $x$ for large values of $x$.

This can also be seen by bringing out one of the powers as a factor.

It follows from $f(x)=x^2-4x^{\frac{1}{2}}=-4x^{\frac{1}{2}}\cdot \left(1-\tfrac{1}{4}x^{\frac{3}{2}}\right)$ that for small values of $x$ the term $x^{\frac{3}{2}}$ is very small (at least much smaller than $x^{\frac{1}{2}}$ ) and the expression between brackets is close to $1$. Then $f(x)$ looks like $x\mapsto -4x^{\frac{1}{2}}$.

It follows from $f(x)=x^2-4x^{\frac{1}{2}}=x^2\cdot \left(1-4x^{-\frac{3}{2}}\right)$ that for large values of $x$ the term $x^{-\frac{3}{2}}$ is very small and the expression between brackets is close to $1$. Then $f(x)$ looks like $x\mapsto x^2$.