Final answer:
The secant line becomes the tangent line as the points get closer together. The limit describes the value the tangent line approaches as the points approach each other. The slope of the tangent line represents the instantaneous rate of change at the point.
Step-by-step explanation:
The secant line is a line that intersects a curve at two or more points. When we take two points on a curve and move them closer and closer together, the line they determine is called the tangent line. The tangent line represents the instantaneous rate of change of the curve at that point. It provides an approximation of the curve at the point where the two points are approaching each other.
The limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value. In this context, as we shrink the two points closer and closer together, we are approaching a particular point on the curve, and the limit represents the value that the tangent line is approaching as the distance between the points approaches zero.
The slope represents the steepness of a line. In this context, when we shrink the two points closer and closer together, the slope of the tangent line represents the instantaneous rate of change of the curve at that point. It tells us how the curve is changing at that specific point.