Question

How do you you use limits to find the slope of a tangent line?

Answer 1

The slope of a tangent line is defined using limits.

Explanation:

The slope of the line tangent to the graph of `y=f(x)` at the point `(a,f(a)` can be stated in more than one way, but all involve limits:

It is the limit of the slopes of the secant lines through the point `(a,f(a))` and a second point on the graph as the value of `x` approaches `a` (if the limit exists).

We can say it is the limit of `(Deltay)/(Deltax`) as `Deltax` appraoches `0` -- with on point used for the changes kept constant at `(a,f(a))` (if the limit exists).

We use limit notation to write:

The slope of the line tangent to the graph of `y=f(x)` at the point `(a,f(a)` is:

`lim_(xrarra)(f(x)-f(a))/(x-a)` (if the limit exists).

or

`lim_(hrarr0)(f(a+h)-f(a))/h` (if the limit exists).

or

`lim_(Deltaxrarr0)(f(a+Deltax)-f(a))/(Deltax)` (if the limit exists).

Each definition relies on limits in the same way. Only the notation really differs.

It may helpto look at some of the posts here on Socratic at http://socratic.org/calculus/derivatives/tangent-line-to-a-curve