一.definition and theorem

1. average rate of change

??The average rate of change of y=f(x) with respect to x over the interval [x1,x2] is
$$\frac{\Delta y}{\Delta x}=\frac{f(x_2)?f(x_1)}{x_2?x_1}=\frac{f(x_1+h)?f(x_1)}{h},h=0.$$

2.THEOREM 1—Limit Laws

??If L,M,c,and k are real number and
$$\underset{x\to c}{\lim }f(x)=L$$
??????????and
$$\underset{x\to c}{\lim }g(x)=M$$
??????????then

3.THEOREM 2—Limits of Polynomials

4.THEOREM 3—Limits of Rational Functions

5.THEOREM 4—The Sandwich Theorem

??Suppose that $g(x)\le f(x)\le h(x)$ for all x in some open interval containing c, except possibly at x = c itself. Suppose also that $$\underset{x\to c}{\lim }g(x)=\underset{x\to c}{\lim }h(x)=L$$
??????????then
$$\underset{x\to c}{\lim }f(x)=L$$

5.THEOREM 5

??If $f(x)\le g(x)$for all x in some open interval containing c,except possibly at x = c itself, and the limits of ? and g both exist as x approaches c,then$$\underset{x\to c}{\lim }f(x)\le \underset{x\to c}{\lim }g(x)$$

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