Limits and continuity

Reach infinity within a few seconds! Limits are the most fundamental ingredient of calculus. Learn how they are defined, how they are found (even under extreme conditions!), and how they relate to continuous functions.

A limit is the value that a function or sequence "approaches" as the input or index approaches some value. In this tutorial, we supply an intuitive understanding of limits.

Find limits by creating a table of values and analyzing the trend.

Now that you have an intuitive understanding of limits, let's do what mathematicians do best and define them rigorously! This definition may be hard to grasp at first, but its beauty will get you in the end.

Review your understanding of the basics of limits with some challenge problems.

A function is continuous at a point if its limit at that point exists and is equal to the actual function's value at that point.

Continuous functions are continuous at all of the points in their domains. In essence, these are functions whose graphs can be drawn with a single brush stroke.

The intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. This is a basic but important property of all continuous functions.

Review your understanding of continuity with some challenge problems.

Take your first steps in finding limits algebraically. For example, find the limit of x²+5x at x=2, or determine whether the limit of 2x/(x+1) at x=-1 exists.

There are some limits that want us to work a little before we find them. Learn about two main methods of dealing with such limits: factorization and rationalization. For example, find the limit of (x²-1)/(x-1) at x=1.

The Squeeze theorem (or Sandwich theorem) states that for any three functions f, g, and h, if f(x)≤g(x)≤h(x) for all x-values on an interval except for a single value x=a, and the limits of f and h at x=a are equal to L, then the limit of g at x=a must be equal to L as well. This may seem simple but it's pure genius. Learn how it helps us find tricky limits like sin(x)/x at x=0.

Find limits of trigonometric functions by manipulating the functions (using trigonometric identities) into expressions that are nicer to handle. For example, find the limit of sin(x)/sin(2x) at x=0.

Remember one-sided limits? Well, these are very useful when dealing with piecewise functions. For example, analyze the limit at x=2 of the function that gives (x-2)² for values lower than 2 and 2-x² for values lager than 2.

Removable discontinuities are points where a function isn't continuous but can become continuous with a small adjustment. Analyze such points and determine what adjustments should be made to "remove" them.

Review your limit-evaluation skills with some challenge problems.

Some limits don't approach a specific value, but instead become boundlessly large as they approach the limiting value. For example, the limit of 1/x as we approach x=0 from the right. Learn about this type of limits and how it relates to vertical asymptotes.

Review your understanding of infinite limits with some challenge problems.