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Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 7.

Limits

This section covers limits and some other precalculus topics.

Definition

  • \lim_{x\to n}f(x) is the value that f(x) approaches as x approaches n.
  • \lim_{x\uparrow n}f(x) is the value that f(x) approaches as x approaches n from values of x less than n.
  • \lim_{x\downarrow n}f(x) is the value that f(x) approaches as x approaches n from values of x more than n.
  • If \lim_{x\to n}f(x)=f(n), then f(x) is said to be continuous in n.

Properties

Let f and g be real functions. Then:

  • \lim(f+g)(x)=\lim f(x)+\lim g(x)
  • \lim(f-g)(x)=\lim f(x)-\lim g(x)
  • \lim(f\cdot g)(x)=\lim f(x)\cdot\lim g(x)
  • \lim\left(\frac{f}{g}\right)(x)=\frac{\lim f(x)}{\lim g(x)}

Squeeze Play Theorem (or Sandwich Theorem)

Suppose f(x) is between g(x) and h(x) for all x in the neighborhood of S. If g and h approach some common limit L as x approaches S, then \lim_{x\to S}f(x)=L.


Diverging-Converging Theorem

A series \sum_{i=0}^{\infty}S_i converges iff \lim S_i=0.

Focus Theorem

The statement \lim_{x\to n}f(x)=L is equivalent to: given a positive number \epsilon, there is a positive number \gamma such that 0<|x-n|<\gamma\Rightarrow |f(x)-L|<\epsilon.


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