Limits: The Foundation of Calculus
Limits describe the behavior of a function as it approaches a specific point, often used for finding the rate of change of a function.
Intuitive Understanding of Limits
A limit describes the value that a function approaches as the input (usually x) gets closer and closer to a particular value.
ExampleConsider the function $f(x) = \frac{x^2 - 1}{x - 1}$. As x approaches 1, this function gets closer and closer to 2, even though it's undefined at x = 1. We write this as:
$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$$
Common MistakeThe limit of a function as it approaches a point is not the same as the function at that point. We don't care any bit about what it is at that point—only its behavior while as it approaches that point.
Estimating Limits from Tables and Graphs
Students learn to estimate limits using tables of values and graphs. This approach helps build intuition about function behavior near critical points.
ExampleTo estimate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$, we can create a table:
From this, we can estimate that the limit as x approaches 2 is 4.
TipWhen estimating limits from graphs, look at the y-values the function approaches as x gets very close to the point of interest from both sides.
The Derivative: Gradient Function and Rate of Change
The derivative is a fundamental concept in calculus, representing both the gradient function of a curve and the rate of change of a function.
Derivative as Gradient Function
The derivative at a point gives the slope of the tangent line to the curve at that point, giving us the rate of change of the function at that point. This interpretation allows us to understand how steep a curve is at any given point.
