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

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

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

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

    whence

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

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

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

    Here we have

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

    whence

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

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.

Noah Dana-Picard 2007-12-28