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http://puu.sh/1aihI

In the 2nd graph, is there an asymptote?

Thanks!

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    There is a vertical asymptote $x=2$ in both pictures.2012-10-01
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    are you sure? but there is a solid black point at x=2 in the second graph o-o2012-10-01
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    There's still an asymptote. Solid black point notwithstanding. Also, when someone goes to the trouble of giving you an answer (and a good answer too), don't you think it's a bit rude to ask if he's sure about it?2012-10-01
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    sorry I really didn't mean to be rude, just wasn't sure why there's a black point and it's still an asymptote.. but yea i got now it thanks guys!2012-10-01
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    The fact that the function has a defined value at $x=2$ doesn’t keep $x=2$ from being a vertical asymptote as $x\to 2^-$. Note that $x=2$ is not a vertical asymptote as $x\to 2^+$ in either picture, though for different reasons: in the first picture $x$ **can’t** approach $2$ from the right.2012-10-01
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    It’s okay; I wasn’t offended. In fact it was helpful, since it pinned down what you were unclear about.2012-10-01
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    Also, in this graphic http://puu.sh/1ajGX, This function is discontinuous, but is it continuous from the right, from the left?2012-10-01
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    From what you were saying, it should be neither right?2012-10-01
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    Yes, neither, since a function is only said to be continuous at points where it is defined. The function does have both a left-sided and a right-sided limit, and could be made left-continuous or right-continuous, but not both, by assigning the respective limit as the missing function value.2012-10-01

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The definition of a vertical asymptote is important if you want to understand if something is an asymptote or not. From Calculus by Varberg, Purcell, and Rigdon:

The line $x = c$ is a vertical asymptote of the graph of $y = f(x)$ if any of the following four statements is true.

  1. $\lim\limits_{x \to c^+} f(x) = \infty$
  2. $\lim\limits_{x \to c^+} f(x) = -\infty$
  3. $\lim\limits_{x \to c^-} f(x) = \infty$
  4. $\lim\limits_{x \to c^-} f(x) = -\infty$

That's it. In both graphs you show, we have that statement 3, where $c = 2$, from the definition is true. Since one of those statements is true, the definition says that $x = 2$ is a vertical asymptote in both cases. Notice, the definition doesn't say anything about the value of the function at $x = c$, only the behavior of the graph as $x$ approaches $c$.