# Limits from equations

Contents
Now that we have all the conceptual stuff laid down, we can start have some fun with finding limits of various functions. Some of these limits don't want you to find them so fast, but we're sure you'll get them in the end!
10 exercises available

### Limits from equations (direct substitution)

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.

### Limits from equations (factoring & rationalizing)

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.

### Squeeze theorem

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.

### Limits of trigonometric functions

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.

### Limits of piecewise functions

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

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: Limits from equations

Review your limit-evaluation skills with some challenge problems.