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Definition of tangent line

  1. Mar 15, 2015 #1
    1. The problem statement, all variables and given/known data

    My textbook says that the slope of the tangent line at a point can be expressed as a limit of secant lines:

    m = \underset{x \rightarrow a}{\lim} \, \frac{f(x) - f(a)}{x - a} \, .

    If x > a and we approach a from the right, why do we have to insist that this limit exists? Why can't we settle for the right-handed limit instead?

    2. Relevant equations

    3. The attempt at a solution

    I'm really not sure why the left limit needs to exist. Any help is appreciated.
  2. jcsd
  3. Mar 15, 2015 #2
    Because sometimes the left limit is different from the right limit. Then the limit doesn't exist, and you don't have any one tangent line. Look at the function [tex]f(x) = |x|[/tex] at [itex]x = 0[/itex]. What is the left hand tangent line limit? What's the right one?
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