Infinite Musings

Students get confused about infinity, \infty , because they think of it as a number, because \infty used like a number. Even though they know there is no largest number, they think of infinity as the largest number.

In studying limits, the starting point of calculus, infinite limits come up early on. We tell them \displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{1}{{{{x}^{2}}}}=\infty . But what we really mean, and what this symbol means, is that by taking x close enough to 0, \displaystyle\frac{1}{{{{x}^{2}}}} eventually becomes larger that any number they choose, no matter how large.

So, instead why don’t we just say \displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{1}{{{{x}^{2}}}}>M where M is any real number? (And as I always suggest when you see the word “any” replace it with “every” and “all.”

    • \displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{1}{{{{x}^{2}}}}>M where M is any real number
    • \displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{1}{{{{x}^{2}}}}>M where M represents every real number
    • \displaystyle\underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{1}{{{{x}^{2}}}}>M where M is all real numbers

Likewise, \displaystyle \underset{{x\to 2}}{\mathop{{\lim }}}\,\frac{{-1}}{{{{{\left( {x-2} \right)}}^{2}}}}<N where N is any number, instead of saying the limit is -\infty .

Infinity, \infty , is really defined by this idea; it’s what “infinity” means.

Consider, \displaystyle \underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{1}{x} . Here, you cannot say the expression is larger than any number (because of the negative values approaching zero from the left), so obviously there is no limit, the limit does not exist, DNE, and using \infty is wrong).

To answer my own question, we don’t do this because we’d have to change all the calculus books, and that’s not going to happen. So, don’t do it. Maybe you can start with this and then quickly switch over to the shorthand version  \infty .

For more on using \infty and DNE see the post Finding Limits and Good Question 5


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Finding Limits

Ways to find limits (summary):

  1. If the function is continuous at the value x approaches, then substitute that value and the number you get will be the limit.
  2. If the function is not continuous at the value approaches, then
    1. If you get something that is not zero divided by zero, the limit does not exist (DNE) or equals infinity (see below).
    2. If you get \frac{0}{0} or \frac{\infty }{\infty } the limit may exist. Simplify by factoring, or using different trig functions. Later in the year a method called L’Hôpital’s Rule can often be used in this situation.
  3. Dominance is a quick way of finding many limits. Exponentials dominate, polynomials, polynomials dominate logarithms, higher powers dominate lower powers. The next post will give some hints about dominance.

Infinity is not a number, but it often is used as if it were. When we say a limit is infinity, what we mean is that the function increases without bound, or there is some x-value that will make the expression larger than any number you choose. Writing things like \infty -\infty =0,\frac{\infty }{\infty }=1,\infty +\infty =2\infty  are common mistakes.

DNE or Infinity?  \displaystyle \underset{x\to 3}{\mathop{\lim }}\,\frac{1}{{{\left( x-3 \right)}^{2}}} does not exist, and DNE is a correct answer. However, it is a bit better to say the limit is (equals) infinity, indicating that the expression gets larger without bound as x approaches 3. Both answers will get credit on an AP exam.  \displaystyle \underset{x\to 3}{\mathop{\lim }}\,\frac{1}{x-3} DNE since the one-sided limits (from the left and from the right) are different.  Only DNE gets credit here.

Take a look at this AP question 1998 AB-2: In (a) students found that \displaystyle \underset{x\to \infty }{\mathop{\lim }}\,2x{{e}^{2x}}=\infty \text{ or }DNE, in (b) they found the minimum value of 2x{{e}^{2x}}  is -{{e}^{-1}} and in (c) they had to state the range of the function is [-{{e}^{-1}},\infty )\text{ or }x>-{{e}^{-1}}. Thus making the students show they knew that this kind of DNE is the kind where the value increases without bound.