Limits at infinity 2

Definitions

  • Intuitive definition of a limit at infinity
    • Recall the functions \(f(x)=100\cdot2^{-x}\) and \(t(x)=10(1-2^{-x})\) from last time.
    • Checking some values on the graphs, we see that the values approach 0 resp. 10 as \(x\) gets very large.
    • We write this as \(\lim_{x\to\infty}f(x)=0\) and \(\lim_{x\to\infty}t(x)=10\).
    • Intuitive definition of a limit at infinity. Let \(f(x)\) be a function defined on some interval \((a,\infty)\). Then \(\lim_{x\to\infty}f(x)=L\) means that we can make sure that \(f(x)\) is arbitrarily close to \(L\) by taking \(x\) large enough.
  • Limit at negative infinity.
    • Consider now the function \(f(x)=\frac{x^2-1}{x^2+1}\).
    • Checking out its graph, we see that \(f(x)\to1\) as \(x\to\infty\).
    • Moreover, we can see that \(f(x)\) approaches 1 also when \(x\) gets very small.
    • We write the latter as \(\lim_{x\to-\infty}f(x)=1\).
    • Intuitive definition of a limit at minus infinity. Let \(f(x)\) be a function defined on some interval \((-\infty,a)\). Then \(\lim_{x\to-\infty}f(x)=L\) means that we can make sure that \(f(x)\) gets arbitrarily close to \(L\) by taking \(x\) small enough.

Horizontal asymptotes

  • Horizontal asymptotes.
    • Consider first the function \(f(x)=\tan(x)\) with domain \((-\frac{\pi}{2},\frac{\pi}{2})\).
    • We have seen that \(\lim_{x\to(-\pi/2)^+}f(x)=-\infty\), and \(\lim_{x\to(\pi/2)^-}f(x)=\infty\).
    • This is why the lines \(x=-\frac{\pi}{2}\) and \(x=\frac{\pi}{2}\) are vertical asymptotes of the curve \(y=f(x)\).
    • Consider now the inverse function \(f^{-1}(x)=\tan^{-1}(x)\).
    • Correspondingly, we have \(\lim_{x\to-\infty}f^{-1}(x)=-\frac{\pi}{2}\) and \(\lim_{x\to\infty}f^{-1}(x)=\frac{\pi}{2}\).
    • This is why the lines \(y=-\frac{\pi}{2}\) and \(y=\frac{\pi}{2}\) are horizontal asymptotes of the curve \(y=f^{-1}(x)\).
    • Definition. The line \(y=L\) is a horizontal asymptote of the curve \(y=f(x)\), if either \[ \lim_{x\to-\infty}f(x)=L\text{ or }\lim_{x\to\infty}f(x)=L. \]
    • Exercises. 2.6: 11, 12.

Limit laws for limits at infinity

  • We can extend the limit laws to limits at infinity using the following observation.
    • Let \(f(x)=\frac{1}{x}\).
    • Then we can see that \(\lim_{x\to-\infty}f(x)=0\) and \(\lim_{x\to\infty}f(x)=0\).
    • This implies the following.
    • Theorem. (a) Let \(r>0\) be a rational number. Then we have \(\lim_{x\to\infty}x^{-r}=0\). (b) Suppose that \(x^{-r}\) is defined everywhere. Then we have \(\lim_{x\to-\infty}x^{-r}=0\).
    • Exercises. 2.6: 18, 20, 24, 28