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I'm looking for a way to determine a one sided limit algebraically, such as

$$\color{blue}{f(x) = \frac {|x|}{x} , x \neq 0}$$

I know that you can find the limit by plugging in numbers or graphing it, but there must be a way to find it without using either of those as a crutch.

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Use the definition of $|x|$ as a piecewise function. Of course, there is no limit for $|x|/x$ as $x$ approaches zero. – Tyler May 16 '11 at 0:12
thank you! I can't believe I missed that – Caleb Jares May 16 '11 at 0:13
and what about equations such as lim(x->-2 from the left) of 1/(x+2)^2 – Caleb Jares May 16 '11 at 0:16
Well, in that case, the limit diverges to $+\infty$ whether you approach from the left or from the right. If I give you a big number, $N$, you'd be able to find a number greater than that by plugging in a value that is close to (but not equal) to $-2$. Are you using the epsilon-delta definitions of the limit to prove convergence and divergence? – Tyler May 16 '11 at 0:21
@Tyler - Can you give an example of working that out? – Caleb Jares May 16 '11 at 0:35
up vote 3 down vote accepted

Recall that $$|a| = \begin{cases} a, & \mbox{if } a \ge 0 \\ -a, & \mbox{if } a < 0. \end{cases} $$

Using this definition you should be able to use normal limit techniques ($\epsilon-\delta$ or what have you)

Notice, of course, that your limit does not exist as $x$ approaches zero.

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