极限与连续的区别

The difference between the limit of function and continuous function

Definition

limit of function:

• The assertion $\lim_{x \to x_0}f(x)=A$ means that we can insure that the absolute value ${\left | f(x)-A \right |}$ is as small as we please merely by requiring that the absolute value $\left | x - x_0 \right |$ be sufficiently small ,and different from zero. This verbal statement is expressible in terms of inequalities as follows: Suppose $\varepsilon$ is any positive number.Then there is some positive number $\delta$ such that
  $${\left | f(x)-A \right | < \varepsilon }\quad if \quad {0< \left | {x-x_0} \right | < \delta}$$

continuous function:

• Suppose the function f is defined at $x_0$ and for all values of x near $x_0$ .Then the function is said to be continuous at $x_0$ provided that
  

  $$\lim_{x \to x_0}f(x)=f(x_0)$$

  注意区别：

  The limit (if it exists) is the number the function approaches. In precise terms LL is the limit of ff at aa if for every ϵ>0ϵ>0, there is a δ>0δ>0, so that for every xx, 0<|x−a|<δ⟹|f(x)−L|<ϵ0<|x−a|<δ⟹|f(x)−L|<ϵ. When the function is continuous at aa, the number it approaches at aa is f(a)f(a). So in the definition above, LL is replaced by f(a)f(a). You can also change the first part to |x−a|<δ|x−a|<δ since the statement is clearly true for x=ax=a also. That’s why the definitions look so similar.

  简单来说：

  continuity of f(x) at x=c exist only if limit of a function f(x) as x–>c exists and equals to f(c)

极限可能不存在的情况：
1、The limit from right and left exists but are not equal.
举例：
$f(x)=1+\frac{\left | x \right |}{x}$

2、 The values of f(x) may get larger nd larger(tend to infinity) as $x \to x_0$ from one side or the other,or from both sides.
举例：当x趋近于0时，$f(x)=1/x$

3、The values of f(x)may oscillate infinitely often,approaching no limit.
举例：$f(x)=sin(1/x)$ which oscillates infinitely often between -1 and +1 as $x \to 0$ from either side

可能不连续的情况

1、$f(x)$ dose not approach any limit at all as $x \to x_0$

2、it approaches a limit which is different from $f(x)$