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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