Theorem 1: Limit As the Slope of a Function

Our first example of a limit is the slope of a function.

Verbally, the slope of f at a is the limit of the ratio of the change in f(x) to the change in x as the change in x approaches zero. The theorem is seen by simply comparing the definitions of limit and slope. The slope exists exactly when the limit exists; and when they do exist they are equal. Notice that the ratio

is undefined when Δx = 0.

The slope of f at a is also equal to the limit

This is seen by setting

Δx = x - a,

x = a + Δx.

Then when x ≈ a but x ≠ a, we have Δx ≈ 0 but Δx ≠ 0; and

Sometimes a limit can be evaluated by recognizing it as a derivative and using Theorem 1 above.