Learn Online Limits

In the mathematical expression the main concept of limit is used to express a value that a sequence or function approaches as the input or key approaches of some value. The limit is typically reduced as lim as in Lim(xn) = x or represent by the right arrow (→) as in an → a. Let us consider this function f(x) = x2. Examine that as x take values very close to 0, the value of f(x) also move towards 0. We say limits  f(x) = 0  x →0

Rules For how to solve limits


Rule1: In learning online limits, given limits function put x=a .If f(a) is a definite value then

limits  f(x) = f(a)
         x →a

Rule2: In learning online limits, If  proving limits  f(x) is a rational function then factorize the numerator and the denominator.Cancel out the  common factors and then put x=a

Rule3: If the given learning online limits function contains a surd then simplify it by using conjugate surd's.After simplification,put x =a

Rule4: If the given  proving learning online  limits  function contains a series which is capable of being expanded then after making proper expansion and simplifying,cancel the common factors in the numerator and denominator,if any Then, put x =a


Limits Examples


1) Evaluate  proving limits lim     (xm -am ) / (xn -an)
                                             x →a

Solution for proving  limits:  lim    (xm -am /xn -an)  =   lim    {xm -am /x-a) ÷ (xn -an /x-a)}
                                                   x →a                                   x →a
Limits =   lim     (xm -am /x -a)   ÷   lim(xn - an /x -a)
                  x →a                                 x →a
Limits =     (ma n-1) ÷ (nan-1)

Limits =   ma m-1 / na n-1   = (m) /(n a m-n)


2) Evaluate proving  limits lim (x+2)3/3 -  (a +2)3/2 / x-a

                                              x→a

Solution proving  limits:   lim (x+2)3/3 - (a +2)3/2 /  x-a
                                            x→a
                        (x +2)3/2 - (a+2)3/2
=      lim     ------------------------------------------
                (x+2)→(a+2)            (x +2) - (a +2)
          ------------------------------------------------------
=   3/2. (a+2)(3/2 -1) =     3/2(a +2)1/2                      [ lim   (xn -an /x -a)   =  nan-1]
                                                                                       x→a

3) Find Limit (x →2) {3x2-5x+7}

Solution:- Given Limit ( x →2)    {3x2-5x+7}

= 3(2)2-5(2)+7   = 12-10+7 = 9


4) Show that Limit (x →3)  (x2+2x-5)  /  (2x2-5x-1) = 5/2

Solution:-Limit (x →3) (x2+2x-5) / (2x2-5x-1)

= Limit (x →3) (x2+2x-5) /  Limit ( x →3)  (2x2-5x-1)

=[ (3)2+2(3)-5)]  / [ 2(3)2-5(3)+1]  =  (9+ 9 - 5) /  (18-15+1)    = 10/ 4 = 5/2.

We can be solved these practice problems on limits  by learning these limits problems.