Intro to Limits and Continuity for AP Calc AB/BC

By Serena Huang

Concept of Limits: Idea of Getting Closer and Closer

Imagine you're a rabbit and a baby chick has just escaped its den. No matter how fast you run, you never quite catch it. But you do get closer and closer with each step. This idea of getting closer and closer to a point without actually reaching it is the essence of limits. In calculus, a limit represents what happens to a function as we approach a specific point.

Formal definition:

Source: https://calcworkshop.com/limits/finding-limits-graphically/

Basically, as x approaches a constant a, what is the y value of the function f(x) getting closer and closer to?

Sometimes the function will converge on a point, and sometimes it will not.

For example, if we have the graph f(x) = x+1, the limit of x as x approaches 4 of x+1 would = 5.

Source: https://www.desmos.com/calculator

Idea of Continuity vs. Discontinuity: Smooth or Not?

Continuity is all about smooth transitions. A function is continuous at a point x=a if three conditions are met:

1. The function is defined at x=a.

1. The limit of the function as x approaches a exists.

   1. Does the right-hand limit = the left hand limit?

1. The value of the function at x=a equals the limit.

In simpler terms, you can trace a continuous function without lifting your pencil off the paper. No jumps, holes, or abrupt changes – just a smooth, unbroken line.

However, not all functions are continuous everywhere. There are three types of discontinuity:

1. Removable Discontinuity: A hole in the function that can be "filled" to make it continuous.

2. Jump Discontinuity: A sudden jump from one value to another.

3. Infinite Discontinuity: When the function approaches infinity or negative infinity at a point.

Source: https://courses.lumenlearning.com/suny-openstax-calculus1/chapter/continuity/

Interesting side point: https://www.youtube.com/watch?v=EfqVnj-sgcc

You can understand Zeno’s paradox better with an understanding of limits.