Step By Step Calculus » 9.1 - Squeeze and Monotone Theorems

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Synopsis
Given a function, f(x)f(x), we are interested in determining if f(x)f(x) gets closer and closer to some number LL as xx approaches a value aa. If so, we say “the limit of f(x)f(x) as xx approaches aa is LL”, and write \displaystyle \lim_{x\to a} f(x)=L\displaystyle \lim_{x\to a} f(x)=L. In fact we consider any of the following: (i) xx moves forever (unbounded) to the right, which we write as x\to\inftyx\to\infty, (ii) xx moves forever (unbounded) to the left which we write as x\to-\inftyx\to-\infty, (iii) xx approaches some value aa from the left or (iv) xx approaches aa from the right. If these respective limits exist they are denoted \displaystyle \lim_{x\to\infty} f(x), \lim_{x\to-\infty} f(x), \lim_{x\to a^-} f(x), \lim_{x\to a^+} f(x)\displaystyle \lim_{x\to\infty} f(x), \lim_{x\to-\infty} f(x), \lim_{x\to a^-} f(x), \lim_{x\to a^+} f(x) respectively. For simplicity, in some situations we include \displaystyle \lim_{x\to\infty} f(x) \displaystyle \lim_{x\to\infty} f(x) in \displaystyle \lim_{x\to a^-} f(x) \displaystyle \lim_{x\to a^-} f(x) by allowing a=\inftya=\infty.
  • The Monotone Convergence Theorem states that for a function f(x)f(x),
    • If f(x)f(x) is monotone increasing and bounded from above for all values near and to the left of x = ax = a, then the limit of f(x)f(x) exists as xx approaches aa from the left (meaning “from below”). In mathematical notation we write \displaystyle\lim_{x\rightarrow a^-}f(x)\displaystyle\lim_{x\rightarrow a^-}f(x) exists. The limit is in fact the least upper bound (LUB) of the values of f(x)f(x) for ff restricted to any interval of the form (a-\delta, a)(a-\delta, a) for sufficiently small \delta >0 \delta >0.
    • If f(x)f(x) is monotone decreasing and bounded from below for all xx near and to the left of aa, then the limit of f(x)f(x) exists as xx approaches aa from the left. In mathematical notation, \displaystyle\lim_{x\rightarrow a^-}f(x)\displaystyle\lim_{x\rightarrow a^-}f(x) exists, and the limit is the greatest lower bound (GLB) of the range of f(x)f(x) for xx restricted to any interval of the form (a-\delta, a)(a-\delta, a) for sufficiently small \delta >0 \delta >0.
    • If ff is monotonically increasing and bounded from below for all xx near and to the right of aa, then the limit of f(x)f(x) exists as xx approaches aa from the right (meaning “from above”). In mathematical notation, \displaystyle\lim_{x\rightarrow a^+}f(x)\displaystyle\lim_{x\rightarrow a^+}f(x) exists and the limit is the GLB of the range of f(x)f(x) for x\in (a, a+\delta)x\in (a, a+\delta) for any sufficiently small \delta > 0 \delta > 0.
    • If ff is monotonically decreasing and bounded from above for all xx near and to the right of aa, then the limit of f(x)f(x) exists as xx approaches aa from the right. In mathematical notation, \displaystyle\lim_{x\rightarrow a^+}f(x)\displaystyle\lim_{x\rightarrow a^+}f(x) exists and the limit is the LUB of the range of f(x)f(x) for x\in (a, a+\delta)x\in (a, a+\delta) for any sufficiently small \delta > 0 \delta > 0.
  • The Monotone Divergence Theorem covers four symmetric cases for a function f(x)f(x),
    • If ff is monotone increasing without upper bound for all xx near and to the left of aa then ff diverges to +\infty+\infty as x x approaches aa from the left (= from below). In mathematical notation, \displaystyle\lim_{x\rightarrow a^-}f(x)=\infty\displaystyle\lim_{x\rightarrow a^-}f(x)=\infty.
    • If ff is monotone decreasing without lower bound for all xx near and to the left of aa, then ff diverges to -\infty-\infty as xx approaches aa from the left. In mathematical notation, \displaystyle\lim_{x\rightarrow a^-}f(x)=-\infty\displaystyle\lim_{x\rightarrow a^-}f(x)=-\infty.
    • If ff is monotone increasing to the right of aa, and if ff is without lower bound for values of xx near but above aa, then ff diverges to -\infty-\infty as xx approaches aa from the right (= from above). In mathematical notation, \displaystyle\lim_{x\rightarrow a^+}f(x)= -\infty\displaystyle\lim_{x\rightarrow a^+}f(x)= -\infty.
    • If ff is monotone decreasing to the right of aa, without upper bound for values of xx near and to the right of aa, then ff diverges to +\infty+\infty. In mathematical notation, \displaystyle\lim_{x\rightarrow a^+}f(x)=\infty\displaystyle\lim_{x\rightarrow a^+}f(x)=\infty.
    • The limit Does Not Exist (DNE) in each of these cases where the function diverges to +\infty+\infty or -\infty-\infty.
  • The Squeeze Convergence Theorem states that if for a given function f(x)f(x) there are two functions g(x)g(x) and h(x)h(x) such that: g(x)\leq f(x)\leq h(x)g(x)\leq f(x)\leq h(x) for all xx near and to the right of aa, and \displaystyle\lim_{x\to a^+}g(x)=L=\lim_{x\to a^+}h(x)\displaystyle\lim_{x\to a^+}g(x)=L=\lim_{x\to a^+}h(x), then \displaystyle\lim_{x\to a^{+}}f(x)=L\displaystyle\lim_{x\to a^{+}}f(x)=L also. There is a symmetric case for the limit from the left.
  • The Squeeze Divergence Theorem states that if for a given function f(x)f(x) there is a function g(x)g(x) such that: g(x)\leq f(x)g(x)\leq f(x) for all xx near and to the right of aa, and if \displaystyle\lim_{x\to a^{+}}g(x)=\infty\displaystyle\lim_{x\to a^{+}}g(x)=\infty, then \displaystyle\lim_{x\to a^{+}}f(x)=\infty\displaystyle\lim_{x\to a^{+}}f(x)=\infty. There are three other symmetric cases for \displaystyle\lim_{x\to a^{+}}f(x)=-\infty, \displaystyle\lim_{x\to a^{-}}f(x)=\infty\displaystyle\lim_{x\to a^{+}}f(x)=-\infty, \displaystyle\lim_{x\to a^{-}}f(x)=\infty and \displaystyle\lim_{x\to a^{-}}f(x)=-\infty\displaystyle\lim_{x\to a^{-}}f(x)=-\infty.
  • The Unnamed Theorem, a special case of the Squeeze Convergence Theorem, states that if \displaystyle\lim_{x\to a^{+}}\vert f(x)\vert=0\displaystyle\lim_{x\to a^{+}}\vert f(x)\vert=0, then \displaystyle\lim_{x\to a^{+}} f(x)=0\displaystyle\lim_{x\to a^{+}} f(x)=0 as well. Dually, \displaystyle \lim_{x\to a^-} |f(x)|=0\displaystyle \lim_{x\to a^-} |f(x)|=0 implies \displaystyle \lim_{x\to a^-} f(x)=0\displaystyle \lim_{x\to a^-} f(x)=0.
Since we can consider a sequence \{s_n\}_{n=1}^{\infty}\{s_n\}_{n=1}^{\infty} as a function f(n)=s_n ; \; \forall nf(n)=s_n ; \; \forall n with domain D_f=\mathbb{N}D_f=\mathbb{N}, all the theorems presented above for functions apply to sequences as well. Of course for sequences we only consider the limit as nn approaches \infty\infty (i.e., \lim_{n\rightarrow \infty} s_{n}\lim_{n\rightarrow \infty} s_{n}).