Difference of two functions whose limit at some point is positive infinite.

1. Take $f(x)=x^2$ and $g(x)=x^4$ ; then we have

   $$\underset{x \rightarrow +\infty}{\lim} f(x)=+\infty \qquad \underset{x \rightarrow +\infty}{\lim} g(x)=+\infty,$$

   
   
   hence the computation of the limit at $+\infty$ of $f(x)-g(x)$ shows an undeterminate case. We have:

   $$f(x) - g(x)=x^2-x^4= \underbrace{x^4}_{\rightarrow +\infty} \; \u... ...left( \underbrace{\frac {1}{x^2}}_{ \rightarrow 0}-1 \right)}_{\rightarrow -1},$$

   
   
   whence

   $$\underset{x \rightarrow +\infty}{\lim} (x^2-x^4)=-\infty.$$

   
   

2. Take $f(x)=x^4$ and $g(x)=5x^3$ ; then we have

   $$\underset{x \rightarrow +\infty}{\lim} f(x)=+\infty \qquad \underset{x \rightarrow +\infty}{\lim} g(x)=+\infty.$$

   
   
   Here we have

   $$f(x)-g(x)=x^4-5x^3=\underbrace{x^4}_{\rightarrow +\infty} \; \und... ...ace{\left( 1-\underbrace{\frac {5}{x}}_{\rightarrow 0} \right)}_{\rightarrow 1}$$

   
   
   whence

   $$\underset{x \rightarrow +\infty}{\lim} (f(x)-g(x))= +\infty.$$

   
   

Note that we have here an example of what is explained in subsection 5.1. For the difference of two square roots, see subsection 5.3.