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http://puu.sh/1aihI

In the 2nd graph, is there an asymptote?

Thanks!

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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

1 Answers 1

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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 $c$.