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4

THE "LIMIT" INFINITY ()

The definition of "becomes infinite"


INFINITY, the student will see, is not a number, it is not a place -- and it is not a limit. Like any defined word, "infinity" stands for an idea that requires many words. In fact, when we say that the limit of a variable "becomes infinite" -- we mean that it does not approach a limit

DEFINITION 4.1.  "becomes infinite."  If the absolute values of a variable (x or y) become and remain greater than any positive number we might name, however large, then we say that the variable "becomes infinite."

Consider again  y  =   1
x
.

If x becomes a very small number -- if it approaches 0 from the

  right, say -- then y, its reciprocal,  , becomes a very large number.   1
x

will become and remain greater, for example, than 10100000000.  y becomes infinite.

We write, in this case,

Now, although we write "lim," we do not mean it -- because no limit

  exists.  When x approaches 0,   1
x
 does not approach any number, and a

limit is a number.  We write "lim = " as shorthand for saying that there is no limit; the function becomes larger than any number we might name.

If x approaches 0 from the left, then  1
x
 becomes a large negative

number.  We write

When a function becomes infinite in this way as x approaches a value, that indicates the function is discontinuous at that value. (Definition 3.1.)

  Hence the function  y 1
x
 is discontinuous at x = 0.

Whenever a function becomes infinite as x approaches a value a, then the line  x = a  is a vertical asymptote of the graph. (Topic 18 of

  Precalculus.)  The graph of  y 1
x
, then, has a vertical asymptote at x = 0.

Next, let us consider the case when x becomes infinite, that is, when it becomes a large positive number -- when it takes on values to the extreme right of 0.

Then   1
x
 does approach a limit.  It becomes a very small number,

namely 0.  We write

(We should speak in this case of the "limit as x becomes infinite," not as x "approaches infinity."  Because again, infinity is neither a number nor a place.)

Finally, when x becomes infinite negatively, that is, when it assumes

  values to the extreme left of 0 (−), then again  1
x
 approaches 0.  We

write

In other words, whenever x approaches ±∞, the values of  y 1
x

approach the horizontal line  y = 0.   That horizontal line is called a horizontal asymptote of the graph.

   Problem 1.   Evaluate  

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tan x does not exist at  π
2
.  There is no limit.  (Topic 17 and
  Topic 19 of Trigonometry.)  As x approaches  π
2
 from the right, tan x

becomes larger than any number we might name. (Definition 4.1.)


Limits of rational functions

A rational function is a quotient of polynomials  (Topic 6 of Precalculus).  It will have the form

f (x)
g (x)

where f and g are polynomials (g 0).

Apart from the constant term, each term of a polynomial will have a factor xn ().  Therefore let us investigate the following limits.  c could be any constant except 0.

The student should complete each right-hand side as a problem.

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Do it yourself first!

1)  =   0
 
2)  =  
 
3)  =  
 
4)  =  
   Example.   Prove:  

 Solution.   Divide the numerator and denominator by the highest power of x.  In this case, divide them by x²:

According to 1), above, the limit of each term that contains x is 0.  Therefore by the theorems of Topic 2, we have the required answer.

In similar cases, the first step is:  Divide the numerator and denominator by the highest power of x present in either.

   Problem 2.   = 4

On dividing both numerator and denominator by x, the result follows.


   Problem 3.   =

In other words:  When the numerator and denominator are of equal degree,
then the limit as x becomes infinite is equal to the quotient of the leading coefficients.


Problem 4.

   = = = 0
   = =

This problem illustrates:

When the degree of the denominator is greater than the degree of the numerator -- that is, when the denominator dominates -- then the limit as x becomes infinite is 0.  But when the numerator dominates -- when the degree of the numerator is greater -- then the limit as x becomes infinite is .

Change of variable

Consider this limit:

Rather than have the variable approach 0, we sometimes prefer that it become infinite.  In that case, we do a change of variable.  We put

x = 1
z
  or   1
n
 or   1
t
,  it does not matter.  For, x approaching 0 is equivalent to

z becoming infinite.  Then

On replacing x with  1
z
, we let z become infinite.  The limit remains 1.

Where will this come up?  In the limit from which we calculate the number e :

(Lesson 15.)

Problem 5.   In the above limit, change the variable to n, and let it become infinite.

Next Lesson:  The derivative


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