§2.2. Rough definitions of limits

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Presentation transcript:

§2.2. Rough definitions of limits Whether tangent line problems or instantaneous velocity problems, we are dealing with functions. So it is necessary to give the definition of a limit for general functions. Topics: Two-sided limits One-sided limits Infinite limits Vertical asymptotes

Ex 1: Ex 2: Ex 3:

In general, we have three approaches to find the value of a limit: Numerical approach Graphical approach Algebraic approach

x 2 2.5 2.9 2.99 3 3.01 3.1 3.5 4 f(x) 5 5.5 5.9 5.99 ? 6.01 6.1 6.5 10

x 1 ½ 1/3 ¼ 1/5 0.1 0.01 0.001 f(x) 0 0 0 0 0 0 0 0

x 2 2/5 2/9 2/13 2/17 2/21 … 2/(4n+1) ... f(x) 1 1 1 1 1 1 … 1 …

Comparison among these 3 approaches: Numerical approach: no good, tedious, time-consuming, sometimes, it’s misguiding So don’t use this approach unless you are asked to use it. (2)Graphical approach: once you have the graph for the given function, it’s easy to evaluate the limit. But it’s not easy to draw the graph. So it is not recommended. (3) Algebraic approach: the most important approach. Most time we use this one.

Infinite limits

Vertical asymptotes The line x = a is called a vertical asymptote of y=f(x) if at least one of the following statements is true: