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
sin x explicitly as a function of x
-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
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
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
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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 <- e
. Which of the following
statements is true about f where
a. f is continues at x = 2 b. lim )(
does not exist.
c. lim )(
=4 d. lim )(
exist but it is not necessarily 4.
5) "A function : Rf ®is continuous at a point 0
x in R if given e > 0 , there is a
d > 0such that for all x in R with <- d 0
xx we have <- e 0
xfxf )()( which of the
following statements is true in general?
a. e is a small number b. d is a small number
c. d is a function, of 0
x and e d. d is unique
6) A function is a special case of a……… .
a. derivative b. equality c. polynomial d. relation
7) A function f is said to be even if it is defined on a set symmetric with respect to
the ……and if it is possesses the property - = xfxf )()( .
a. origin b. x-axis c. y-axis d. open
8) For any real number x . The …..value of x , denoted by x .
a. absorbency b. absorption c. abstraction d. absolute
9) For a real function f, the …..of f is the set of all pairs yx ),( in R´ R such that
= xfy )( and x is in the domain of the function.
a. curve b. graph c. greatest d. divisor
10. The graph = xgy )( is an odd function has the ….as a line symmetry.
a. y-axis b. origin c. y=x d. x-axis
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1)a 2)b 3)a 4)c 5)c 6)d 7)c 8)d 9)b 10)d