Notes, etc.

The "Limit Theorems" in this section are the key to translating the very difficult "epsilon-delta" definition to some practical rules for calculating limits.

You don't actually need to memorize the theorems, but just the upshot of the conclusions, which is:

1) If a rational, root, or transcendental function is defined at a point, then the value of the limit is the value of the function at the point.

2) The only places where you need to be concerned about multi-part functions (such as absolute value or greatest integer functions) are where the rules change.

3) If you have a rational, root, or transcendental function, and you can algebraically cancel out the divide by zero problem, then the value of the limit is the value you get when you plug in the number, after canceling out the divide by zero problem.

4) When you mix algebraic with transcendental (sine, cosine, logarithm, exponent) functions, you sometimes need to resort to indirect analytical methods, such as the squeeze theorem.

A) See if there's a divide by zero problem. If not, just plug in the number. That's the limit.

B) If there is a problem, see if you can use algebra to cancel out the problem, then plug in the number. If you get a unique answer, that's the limit.

The 4 algebra tricks you'll need are:

i) Factoring.

ii) Simplifying multi-story fractions, then factoring.

iii) Multiplying by the conjugate (like in the square root example, above), then factoring.

iv) Eliminating the absolute value signs, then applying tricks i) - iii).

C) If you can't eliminate the problem, or if you do not get a unique answer, then the limit does not exist.

Here are the 5 examples you'll need to think about, as you calculate limits using algebra: