The limit "infinity" - An approach to calculus

An Approach

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THE LIMIT "INFINITY" (∞)

The definition of "becomes infinite"

Limits of rational functions

Change of variable

INFINITY, along with its symbol ∞, is not a number and it is not a place. To become "infinite" is the mathematical idiom we use to describe a property of a variable under a very specific condition. It is the following.

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."

And so, when we write

we mean: No matter what large number M we might name, as x approaches a, the values of f(x) ultimately become and remain greater than M.

We employ the symbol ∞ in an algebraic statement to signify that the condition — the "If" clause — of Definition 4.1 has been satisfied.
That symbol by itself has no meaning.

As an example, here is the graph of the function

1
x

Let us see what happens to the values of y as x approaches 0 from the right:

As the sequence of values of x become very small numbers, then the sequence of values of y, the reciprocals, , become very large numbers. The values of y will become and remain greater, for example, than 10^100000000 y becomes infinite.

We write, in this case,

When a limit is infinite, as in this case, then even though the limit is not a number, we say nevertheless that the limit "exists." Saying that means that the definition of "becomes infinite" has been satisfied.

If x approaches 0 from the left, then the values of

1
x

become large

negative numbers. In this case, we write

That limit also exists. However, since it is different from the right-hand limit, the "limit as x approaches 0" does not exist Definition 2.2.

When a function becomes infinite as x approaches a value a, the straight 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.

In that case,

1
x

becomes a very small number, namely 0. We write

We should read that as "the limit as x becomes infinite," not as "x approaches infinity," because again, infinity is neither a number nor a place. On the other hand, we could read that however we please ("the limit as x becomes dizzy"), as long as whatever expression we use refers to the condition of Definition 4.1!

See First Principles of Euclid's Elements, Commentary on the Definitions. See especially that a definition is nominal; it asserts only how a word or a name will be used; and we must agree to that.

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 becomes infinite positively or negatively,

the values of y =

1
x

approach the horizontal line y = 0. That line is

called a horizontal asymptote of the graph.

Problem 1. Evaluate

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π
2

, tan x does not exist. (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 (n ≥ 1). Therefore let us investigate the following limits. c could be any positive constant.

As a problem, the student should complete each right-hand side.

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1)		=	0

2a)		=	∞
	n even.

2b)		=	∞
	n odd.
2c)		=	−∞
	n odd.

Compare y =

1
x

above, where n = 1.

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 power of x that appears in the leading term of either one.

Problem 2.

The result follows on dividing both numerator and denominator by x.

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.

In the following, the rational function is the reciprocal of the one above:

∞

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 .