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Here I have I have a perfect square trinomial:

$x^2 - 6x + 9$

Here I have a perfect square binomial:

$x^2 - 9$

I am asked to find the limit as x approaches 3 of $\frac{x^2 - 6x + 9}{x^2 - 9}$. The full equation is illustrated below:

$\lim \limits_{x \to 3} \frac{x^2 - 6x + 9}{x^2 - 9}$

If we plug 3 in for x, you'll notice the result is $\frac{0}{0}$. In other words, the value is undefined and that's typically how it is with these limit problems. We now need to find the values approaching 3 from both the negative and positive end.

But this is where it gets weird for me. The polynomials can be simplified through factoring.

$x^2 - 6x + 9$ can be factored into $(x - 3)(x - 3)$

$x^2 - 9$ can be factored into $(x + 3)(x - 3)$

Since there is a $(x - 3)$ in the numerator and denominator, we can cancel them out to get the following equation:

$\frac{x - 3}{x + 3}$

That means the limit can be rewritten as:

$\lim \limits_{x \to 3} \frac{x - 3}{x + 3}$

But now if we plug in 3, we get $\frac{0}{6}$ which is 0.

How is this possible? How does rewriting the equation give a different answer? More importantly, how does simply rewriting the question make an undefined value suddenly defined?

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  • $\begingroup$ Evaluating limits by substitution is allowed iff the function is continuous at that value (i.e. $\lim_{x\to a}f(x)=f(a)\iff f(x)\ \text{continuous at}\ x=a$). $\endgroup$ – Andrew Chin Apr 11 '20 at 23:54
  • $\begingroup$ @AndrewChin I have no clue what that double sided white arrow with a black order is. Can you rephrase that? $\endgroup$ – Donato Apr 11 '20 at 23:56
  • $\begingroup$ The first sentence is what the math statement says. $\endgroup$ – Andrew Chin Apr 11 '20 at 23:57
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    $\begingroup$ $f(x)=\frac{x^2-6x+9}{x^2-9}$ is not continuous at $x=3$, so you will not be able to evaluate the limit by substitution. However, $g(x)=\frac{x-3}{x+3}$ is continuous at $x=3$, so you can evaluate the limit by substitution $g(3)=0$. $\endgroup$ – Andrew Chin Apr 11 '20 at 23:59
  • $\begingroup$ See this answer. $\endgroup$ – Paramanand Singh Apr 12 '20 at 8:21
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That's a great question. When you take a limit as $x\to 3$, you are concerned with all points around $x=3$, but not actually $x=3$. In any textbook, this is the standard meaning of a limit. By defining it in this way, it allows you to consider the behavior near points.

When you plugged in and got $0/0$, that's an indication that if the limit exists, the function has a hole there. You can always fill in the hole by defining $f(3)=L$, where $L$ is the limit.

Now why did it work out? It's because when you factored and got $\frac{(x-3)(x-3)}{(x-3)(x+3)}$, the reason you were able to cancel the $x-3$ terms is because of what I discussed above. For $x$ close to $3$, but not equal to it, we have $x-3 \neq 0$. Hence, you have a fraction with a common nonzero term on the numerator and denominator so you can cancel them.

When you take the limit of $\frac{x-3}{x+3}$, again you are concerned with all points near $x=3$. It turns out though, the function $f(x)=\frac{x-3}{x+3}$ is continuous at $x=3$ so you are really just observing that $\displaystyle f(3)=\lim_{x\to 3} f(x)=0$. There are no contradictions; you are always plugging in numbers close to $3$, and it's just a coincidence that this was allowed in the second case.

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  • $\begingroup$ And by continuous, you simply mean there is no hole in the curve or line plotted on the cartesian plane. $\endgroup$ – Donato Apr 12 '20 at 0:59
  • $\begingroup$ Yes. To see an illustrative example, let $f(x)=x$ when $x\neq 1$ and $f(x)=2$ for $x=1$. Then $\lim_{x\to 1}f(x)=\lim_{x\to 1} x=1$ (since we are not considering $x=1$ case). However, $f(1)=2$ so the function has a removable discontinuity. $\endgroup$ – zugzug Apr 12 '20 at 1:06
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When you get 0/0, you cannot conclude that the limit is undefined. 0/0 is an "indeterminate form", which means it tells you nothing about the limit. When you get 0/0, the limit could be defined or undefined. You must find a different approach, as you did in the second approach.

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