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  Season 1:Function and Limit
An equation of the form y=f(x) is said to define y explicitly as a function of x (the
function being f), and an equation of the form x=g(y) is said to define x explicitly as a
function of y (the function being g). For example, y=5x
2
sin x explicitly as a function of x
and x=(7y
3
-2y)2/3 defines x explicitly as a function of y.
An equation the is not of the form y=f(x) but whose graph in the xy-plane passes the
vertical line test is said to x, and an equation that is not of the form x=g(y) but whose
graph in the xy-plane passes the horizontal line test is said to define x implicitly as a
function of y.
In the preceding sections we treated limits informally, interpreting
®ax
lim f(x)=L to mean

that the values of f(x) approaches L as x approaches a from either side (but remains 
different from a). However, the phrases 'f(x) approaches L' and 'x approaches a' are 

  intuitive ideas without precise mathematical definitions. This means that if we pick any

positive number, say e , and construct an open interval on they y-axis that extends e

Then is deducing these limits results from the fact that for each of them the numerator 
and denominator both approach zero as h ® 0. As a result, there are two conflicting 
influences on the ratio. The numerator approaching 0 drives the magnitude of the ratio 
toward zero, while the denominator approaching 0 drives the magnitude of the ratio 
toward + ¥ . The precise way in which these influences offset on another determines 

   whether the limit exists and what its value is

In a limit problem where the numerator and denominator both approach zero, it is 
sometimes possible to circumvent the difficulty by using algebraic manipulations to write 
the limit in a different from. However, if that is not possible, as here, other methods are 
required. One such method is to obtain the limit by 'squeezing' the function between 
simpler functions whose limits are known. For example, suppose that we are unable to 
show that 
®ax
lim f(x)=L directly, but we are able to find two functions, g and h, that have 
same limit L as x®a and such that f is 'squeezing' between g and h by means of the 
inequalities g(x) £f(x) £h(x) it is evident geometrically that f(x) must also approach L as 
x®a because the graph of f lies between the graphs of g and h. 
This idea is formalized in the following theorem, which is called the Squeezing Theorem
or sometimes the Pinching Theorem

تست های فصل اول 
1) If the domain of a real-valued, continuous function is connected, then the range is
a. An interval of R it self b. An open set 
c. A compact set- d. A bounded set 
2) A function : ® RAf is said to ……….on A if there exists a constant M > 0 such 
that )( £ Mxf for all Î Ax .
a. be closed b. be bounded 
c. have extremum d. have maximum 
3) A set Í RU is said to be open if for each ÎUx there is ….number a e such that 
-e + e ),( ÍUxx .
a. A positive real b. a non-zero real 
c. complex d. a negative set 
4) Let e > 0 , then it is easy to see that