Example: limit (as x approaches some stated value which we shall refer to here as "a")

Step one is always to plug in the value "a". If you get anything other than the four unacceptables (0/0, 0/infinity, infinity/0 or infinity/infinity), you're finished.

If you get any of the "unacceptables", then the techniques vary depending on the type of problem:

1st choice) try to factor and cancel, then try to plug "a" into the simplified function.

2nd choice) if you have a square root plus-or-minus something (in the numerator or denominator), then multiply top AND bottom by the square root minus-or-plus something (this is the conjugate of the radical). Usually, the radical will move from the numerator to the denominator (or vice-versa) and there will be something that you can cancel. Then plug "a".

3rd choice) if you have a rational expression (no radical and nothing factors), then factor out of the numerator AND the denominator an amount equal to the highest power of the variable. That is, given (x-squared plus x minus 2) over something, factor out the x-squared to get x-squared times (1 plus 1overx minus 2overx-squared). Then, as x approaches infinity (if that is the limit), recall that 1overx is 1 divided by infinity which is (as a limit) zero.

As to how to differentiate, repeated use of the limit definition method of finding the derivative has led us to the shortcut referred to as the general power rule: (something) raised to the nth power will have as its derivative an amount equal to n times (something) raised to the (n-1) power and times the derivative of the (something). I have also referred to this as the derivative from the outside times the derivative of the inside. Thus, the derivative of (2x+3)cubed is 3 times (2x+3)squared times the derivative of (2x+3). 3(2x+3)squared times 2 = 6(2x+3)squared.

Also, check out this interactive Internet site: http://fourier.math.temple.edu/cgi-bin/manager