How do you find the slope of the tangent line using the formal definition of a limit?

You would have to use the equation

#f'(x) =lim_(h->0)(f(x+h)-f(x))/h#

at a given point.

This derives from the average rate of change; as a line gets closer to a point on a graph, it becomes only one intersection of the line and not two (which is a secant line).

Here is a link for graphical examples:

http://facultypages.morris.umn.edu/~mcquarrb/teachingarchive/Precalculus/Lectures/AverageRateofChange.pdf

This is useful in determining the instantaneous rate of change of any curve. For example, if you have #f(x) = x^2#, then you can use the equation above to find the slope of the line at any x-value (by simplification and canceling the h on the denominator). Or, if they specify the value of x, then you just insert the number for x and solve for the problem.

It works for every polynomial equation, but be aware that other equations maybe complicated or may not work.

Later on (if you are currently taking Calculus I), you will learn about derivatives and find ways to easily get the slope of tangent lines; for the #x^2# part, 2x is the slope of the tangent line at any x-value. If you use the laws of derivatives and the limit definition, it turns out to be the same!

Once you understand how limits work in relation to derivatives, it becomes very interesting!