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I am trying to solve $\displaystyle\lim_{x\to3^-}\frac{3}{x - 3} - \frac{3}{x} - 9$. Here's what I have tried.

$ \lim_{x\to3^-}\frac{3}{x - 3} - \frac{3}{x} - 9 \\ \lim_{x\to3^-}\frac{3x}{x(x - 3)} - \frac{3(x-3)}{x(x-3)} - \frac{9x(x-3)}{x(x-3)} \\ \lim_{x\to3^-}\frac{3x - 3x - 9x^2 + 27x}{x(x - 3)} \\ \lim_{x\to3^-}\frac{- 9x^2 + 27x}{x^2 - 3x)} \\ $

By L' Hopital's Rule,

$ \lim_{x\to3^-}\frac{-18x + 27}{2x - 3} \\ \lim_{x\to3^-}\frac{-18}{2} \\ -9 \\ $

However, Wolfram Alpha claims that the limit is infinity: http://www.wolframalpha.com/input/?i=lim+x+-%3E+3+%5Cfrac%7B3%7D%7Bx+-+3%7D+-+%5Cfrac%7B3%7D%7Bx%7D+-+9

Why?

  • 1
    Well, in a sense it can. The conventional answer is that the limit is $-\infty$, since the term $\frac{3}{x-3}$ is less than any given large negative number, provided $x$ is suffiently close to but less than 3.2012-09-22

5 Answers 5

2

In your second step, $3(x-3)=3x-9$, not $3x$. When you restore the missing term, you’ll find that l’Hospital’s rule no longer applies.

1

You made an algebra mistake; instead of

$\lim_{x\to3^-}\frac{3x - 3x - 9x^2 + 27x}{x(x - 3)}$

It is

$\lim_{x\to3^-}\frac{3x - 3x + 9 - 9x^2 + 27x}{x(x - 3)}$

So you just forgot a term.

1

You missed a term $+9$ on the numerator of your fraction. This matters because then the limit is not of the form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$, which is a hypothesis of l'Hôpital's rule.

0

The numerator and denominator of the fraction will both be divisible by $x-3$ if the limit exists. You'll have $ \lim_{x\to3} \frac{(x-3)(\cdots\cdots\cdots)}{(x-3)(\cdots\cdots)}. $ Then cancel the common factor. After that, you can plug in $3$ for $x$.

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Why not use WolframAlpha to help you find where the problem occurred? Your third line already evaluates to -9, so you know the error is not in the rest of your computation.

In fact the plot shows that your third line is identically equal to -9, which hints that you did something to incorrectly cancel out the $3/(x-3)$ and the $3/x$. I'm being deliberately vague here to highlight the fact that this basic level of information is in your grasp.