Why do we need the limit to exist for the slope of the tangent line?

[bigplanet401]

Homework Statement

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?

Homework Equations

The Attempt at a Solution

I'm really not sure why the left limit needs to exist. Any help is appreciated.

 

[axmls]

Because sometimes the left limit is different from the right limit. Then the limit doesn't exist, and you don't have anyone tangent line. Look at the function $$f(x) = |x|$$ at $x = 0$. What is the left hand tangent line limit? What's the right one?