Question

How do you use the limit definition to find the slope of the tangent line to the graph `f(x)=x^3`?

Answer 1

Expand, reduce and evaluate the limit.

Explanation:

Definition 1
If you are using as your definition of the slope at `x`:
`lim_(hrarr0)(f(x+h)-f(x))/h`,

then you'll need to expand `(x+h)^3 = x^3+3x^2h+3xh^2+h^3`.

(If using the slope at `a` rather than at `x`, simply replace the `x`'s above with `a`'s.)

Definition 2
If you are using the slope at `a` defined by

`lim_(xrarra)(f(x)-f(a))/(x-a)`,

then you'll need to factor `x^3-a^3 = (x-a)(x^2+2ax+a^2)` (difference of cubes).