I'm working on my calculus homework right now and have become stumped on two questions about limits.

  1. $\lim\limits_{x \to 2^-} \frac{3x^2 + x - 7}{|x - 2|}$
  2. $\lim\limits_{x \to 0^+} \frac{x}{|x|}$

For the first part, plugging in doesn't work as it will give you 7/0. And you cannot factor it, so I'm at a complete loss.

As for the second question I didn't get very far either, as just plugging in would give you 0/0.

Can someone help me out with this? Also, I'm sorry if this isn't in the proper format for posting equations and if it's incorrect I'd appreciate it if someone could edit it for me.

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  • $\begingroup$ Just to clarify, for 1 should the expression be $3x^2+x-\frac{7}{|x-2|}$ or $\frac{3x^2+x-7}{|x-2|}$? $\endgroup$ – Alex Becker Jan 25 '13 at 1:56
  • $\begingroup$ Limits can be infinite, and not defined. Did you discuss this in class? $\endgroup$ – gnometorule Jan 25 '13 at 1:57
  • $\begingroup$ Alex, the latter. Also I believe we have, I don't remember off of the top of my head though. $\endgroup$ – Matthew Brzezinski Jan 25 '13 at 1:58
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    $\begingroup$ @user1327636 The value of the limit at a point $c$ has nothing to do with the value of the expression at $c$. These are only the same if the expression is continuous at $c$ (this is the definition of continuity, in fact). $\endgroup$ – Alex Becker Jan 25 '13 at 2:04

Since, in first expression, plugging the value gives result $\frac{7}{0}$ which shows the limit is infinite.

For second part,

$\lim_{x\to0^+}\frac{x}{|x|}=\lim_{x\to0^+}\frac{x}{x}=1$ (I have taken $|x|=x$ as x is approaching $0$ from positive side)

you can read from here : http://www.themathpage.com/acalc/infinity.htm

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  • $\begingroup$ The expression should be $\frac{x}{|x|}$, not that it makes a difference. $\endgroup$ – Alex Becker Jan 25 '13 at 2:06
  • $\begingroup$ @AlexBecker: thanks :) $\endgroup$ – Aang Jan 25 '13 at 2:07

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