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I have this limit:

$\lim_{x \to 1} \frac{2x^2-x-6}{x(x-1)^3}$

This can be written as:

$\lim_{x \to 1^+} \approx \frac{-5}{1\times\mbox{tiny positive}} \to - \infty$

Why is that? I mean, let's plug in some numbers. $-5/1.0000000001$ is almost $-5$ , the greater the denominator becomes the close the number to $-5$

Can anybody tell me why the book says it goes to -infinity? Thanks a lot

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    I made the change but my edit needs to be OP approved.2011-11-11

1 Answers 1

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You didn't choose a tiny positive number! Rather than $1.00000000001$, you should have chosen $0.0000000000001$


One benefit of choosing such tiny decimal numbers is that we can represent them as fractions. So we would get fractions $\frac{1}{1000} , \frac{1}{10000000} , \frac{1}{100000000000} $ ,etc.

But dividing by these fractions is the same as multiplying by $1000, 10000000, 100000000000$, etc. The effect of this is that the fraction grows arbitrarily large.

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    Glad to help! .2011-11-11