1.5, Limits and infinity

Today we look at limits involving infinity.  In the first half, we look at the meaning of the input variable, x going to infinity.  In the second half, we examine the meaning of the output variable, y, going to infinity.

Limits as x goes to infinity

Infinity is a mathematical concept, not a number. In this class, we will view (positive) infinity as "larger than any real number" or "beyond all positive real numbers." The limit of f(x), as x goes to infinity, is the value that f(x) approaches, as x grows beyond bound.

For example, consider the rational function f(x) = (4x^2+5x-1)/(3x^2-7x).  What is the limit of f(x) as x goes to infinity.  Consider plugging in big numbers like x equals one billion (1,000,000,000). Here is f(1,000,000,000).

From this computation, it appears that the limit of this function, as x goes to a billion, 1.3333... .

We can build up some intuition about this.  As x gets large, 4x^2 begins to dominate the numerator.  If, for example, x is a billion then x^2 is a billion billion, much larger than 5x which is a mere five billion. So, intuitively, one might argue that only the higher power sof x matter.  If so then we could argue that

Solving limits by algebra

A fundamental example (and major result that we need) is the limit, as x goes to infinity, of 1/x.

As x grows beyond bound, passing a million, billion, ... googol (10^100), ..., googleplex (10^googol), ... and so on, the quantity 1/x becomes smaller and smaller, approaching zero.  If x = 10^n, that is, if x in decimal notation is a 1 followed by n zeroes, then 1/x = 10^(-n), so 1/x, in decimal notation, is a decimal point followed by n-1 zeroes and then a 1.

We summarize this by saying that the limit, as x goes to infinity, of 1/x is 0.

This limit is useful in algebraically simplifying other limits.

For example we could reconsider the function f(x) = (4x^2+5x-1)/(3x^2-7x) mentioned above. Multiplying both numerator and denominator by 1/x^2, we can rewrite the rational function as

(This algebra is correct as long as x is not zero.  Since we are trying to compute that limit as x gets very large, we clearly do not have to concern ourselves with x being zero!)

Since 1/x goes to zero as x goes to infinity, then expressions like 5/x or 1/x^2 or 7/x will go to zero also.  So our limit can be algebraically manipulated as follows:

Limits and vertical asymptotes

Consider the graph, below, of a rational function.  The function is drawn in blue. A vertical asymptote, x = -3,  is drawn in red and a horizontal asymptote, y = 1, is drawn in green.

In this case, as x approaches -3 from the left, the y values shoot up beyond bound. So here we have infinity as a limit!  The output is growing beyond bound!

On the other hand, as x approaches -3 from the right, the y values shoot down out of sight, dropping below every real number.  In this case we say negative infinity is the limit.  Our notation for this would be

The green horizontal asymptote tells us what happens as x goes off to the right, beyond all positive real x-values and also tells us what happens as x goes off to the left, beyond all negative x values. In either case, y is approaching 1 and so our limit is 1.

The vertical asymptotes give us limits of infinity while the horizontal asymptotes describe y values as x goes to infinity.

Resources

Infinite limits are covered in section 2.2 of the OpenStax textbook.