Step By Step Calculus » 9.2 - Definition of Limits and Asymptotes

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Synopsis
There are several reasons we need to carry out an in depth study of limits. Among the most important:
  • To predict what will eventually happen to some real world process modeled by a function.
  • To determine what happens as we successively improve approximations to a function.
  • To determine the behavior of a function near a point where the function is perhaps undefined, or it is difficult to compute or even approximate.
  • To find the slope of the tangent line to a curve, as we described it only for the tangent line to a parabola in the section of “Lines and Quadratics”.
Convergence as xx approaches infinity:
  • ff converges to LL as x\to\inftyx\to\infty if given any \epsilon>0\epsilon>0 there exists an N>0N>0 (which can depend upon \epsilon\epsilon) such that
    \displaystyle x\geq N \; \Rightarrow \; |f(x)-L|<\epsilon \; .
    \displaystyle x\geq N \; \Rightarrow \; |f(x)-L|<\epsilon \; .
    This is written \displaystyle \lim_{x\to \infty} f(x)=L \displaystyle \lim_{x\to \infty} f(x)=L.
    The only way this can fail is if you specify a particular \epsilon\epsilon and can not find any NN that will work for that value of \epsilon\epsilon. This means that no matter how far out I go (i.e., choose NN) there is at least one value of xx further on ( x > Nx > N) for which f(x)f(x) is separated from LL by more than that value of \epsilon\epsilon :
    \displaystyle \exists \epsilon >0 \; \displaystyle \exists \epsilon >0 \; such that \displaystyle ; \forall N\in \mathbb{N} \; \exists x > N \; \displaystyle ; \forall N\in \mathbb{N} \; \exists x > N \; with\displaystyle |f(x) - L | > \epsilon \displaystyle |f(x) - L | > \epsilon
    This mathematical statement represents the concept of \epsilon\epsilon-separated:
    For a number \epsilon\epsilon, ff is \epsilon\epsilon-separated from LL as x\to\inftyx\to\infty if for every N>0N>0 we have |f(x)-L|\geq \epsilon|f(x)-L|\geq \epsilon for somex > Nx > N. Then the precise definition becomes: ff converges to LL as x\to\inftyx\to\infty if ff is not \epsilon\epsilon-separated from LL given any \epsilon>0\epsilon>0.
  • Dually, ff converges to LL as x\to-\inftyx\to-\infty if given any \epsilon>0\epsilon>0 there exists an N<0N<0 (which can depend upon \epsilon\epsilon) such that
    \displaystyle |f(x)-L|<\epsilon \qquad \forall x< N.
    \displaystyle |f(x)-L|<\epsilon \qquad \forall x< N.
    This is written \displaystyle \lim_{x\to -\infty} f(x)=L \displaystyle \lim_{x\to -\infty} f(x)=L.
If \displaystyle\lim_{x\rightarrow\infty }f(x)\displaystyle\lim_{x\rightarrow\infty }f(x) or \displaystyle\lim_{x\rightarrow -\infty }f(x)\displaystyle\lim_{x\rightarrow -\infty }f(x) exists and is equal to LL, then ff has a horizontal asymptote at y = L.y = L.
Convergence as xx approaches a number:
When we are dealing with approach to a finite value x \to a x \to a, then instead of thinking of xx as being beyond a certain NN, we have to specify that xx needs to be close to aa, and we may only be interested in approaching aa from one side:
  • The function ff converges to LL as x\to ax\to a from the right if for any \epsilon>0\epsilon>0 there exist a \delta>0\delta>0 such that
    \displaystyle a<x<a+\delta \;\displaystyle a<x<a+\delta \; and \displaystyle \; x\in D_f\; \Rightarrow \; |f(x)-L|<\epsilon \displaystyle \; x\in D_f\; \Rightarrow \; |f(x)-L|<\epsilon
    This is written \displaystyle \lim_{x\to a^+} f(x)=L \displaystyle \lim_{x\to a^+} f(x)=L.
  • Dually, the function ff converges to LL as x\to ax\to a from the left if for any \epsilon>0\epsilon>0 there exist a \delta>0\delta>0 such that
    \displaystyle a-\delta<x<a \; \Rightarrow \; |f(x)-L|<\epsilon.
    \displaystyle a-\delta<x<a \; \Rightarrow \; |f(x)-L|<\epsilon.
    This is written \displaystyle \lim_{x\to a^-} f(x)=L\displaystyle \lim_{x\to a^-} f(x)=L.
If \displaystyle\lim_{x\rightarrow a^+}f(x)=\displaystyle\lim_{x\rightarrow a^-}f(x)=L\displaystyle\lim_{x\rightarrow a^+}f(x)=\displaystyle\lim_{x\rightarrow a^-}f(x)=L, then we say \displaystyle\lim_{x\rightarrow a}f(x)=L\displaystyle\lim_{x\rightarrow a}f(x)=L. If one (or both) of the one-sided limits does not exist, or they both exist but do not approach the same value, then we say \displaystyle\lim_{x\rightarrow a}f(x) \displaystyle\lim_{x\rightarrow a}f(x) does not exist for x\to ax\to a.
Divergence at infinity:
  • A function ff diverges to \infty\infty (or -\infty-\infty) as x\to\inftyx\to\infty if for any M>0M>0 there is an N_MN_M such that
    \displaystyle x > N_M \; \Rightarrow \; f(x)>M \; \displaystyle x > N_M \; \Rightarrow \; f(x)>M \; (or f(x)<-Mf(x)<-M respectively)\displaystyle \; .\displaystyle \; .
  • A function ff diverges to \infty\infty (or -\infty-\infty) as x\to-\inftyx\to-\infty if for any M>0M>0 there is an N_MN_M such that
    \displaystyle x < N_M \; \Rightarrow \; f(x) > M \; \displaystyle x < N_M \; \Rightarrow \; f(x) > M \; (or \displaystyle \; f(x)<-M \; \displaystyle \; f(x)<-M \; respectively)\displaystyle \displaystyle
Divergence at a number:
  • A function f(x)f(x) diverges to \infty \infty (or -\infty -\infty ) as x\to ax\to a from the right if for any M>0M>0 there is a \delta>0\delta>0 such that f(x)>Mf(x)>M(or f(x)<-Mf(x)<-M respectively) for all x\in(a,a+\delta)x\in(a,a+\delta). This is written as \displaystyle \lim_{x\to a^+}f(x)=\infty\displaystyle \lim_{x\to a^+}f(x)=\infty (or -\infty-\infty).
  • A function f(x)f(x) diverges to \infty \infty (or -\infty -\infty ) as x\to ax\to a from the left if for any M>0M>0 there is a \delta>0\delta>0 such that f(x)>Mf(x)>M(or f(x)<-Mf(x)<-M respectively) for all x\in(a-\delta,a)x\in(a-\delta,a). This is written as \displaystyle \lim_{x\to a^-}f(x)=\infty\displaystyle \lim_{x\to a^-}f(x)=\infty (or -\infty-\infty).
ff has a vertical asymptote at aa if either \displaystyle \lim_{x\to a^+} f(x)=\pm\infty\displaystyle \lim_{x\to a^+} f(x)=\pm\infty or \displaystyle \lim_{x\to a^-}f(x)=\pm\infty.\displaystyle \lim_{x\to a^-}f(x)=\pm\infty.