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, the student will see, is not a number and it is not a place. 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

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 10^100000000. 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

Whenever a function becomes infinite, as in this example, 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 (n ≥ 1). Therefore let us investigate the following limits. c could be any constant except 0.

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

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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 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 .