Limits and Rates of Change

The Tangent Problem

A tangent line is a line touches the graph of a function exactly once in a region. This means that, within an arbitrarily small circle, the line intersects the graph of the function at one point only. The slope of the tangent line is rate of change of y with respect to x at that point. This can also be stated as the instantaneous rate of change of the slope of the line.

The Velocity Problem

Drop a ball from shoulder height. It is clear that the ball does not travel the same speed as it falls, but speeds up. What is the velocity (speed) of the ball at any particular moment? We can take an average velocity by measuring the velocity at two points, but this does not give us an exact velocity. The velocity of the ball at an exact moment in time is called the instantaneous velocity.

The Limit of a Function

We write Limit( f(x), x → a ) = L if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a on either side of a, but not equal to a.

The definition of a limit implies that the f(x) is 'well-behaved' near x = a. Take, for example, the function f(x) = sin( π / x ) near x = 0. As x gets closer to zero, sin( π / x ) begins oscillating rapidly, and does not approach any certain value. We can say that Limit( sin( π / x ), x → 0 ) does not exist.