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Examine the following functions for continuity. $f(x) = \frac {1} { x - 5}, x \: \neq \: 5 $

$\begin{array}{1 1} \text{Yes, it is continuous} \\ \text{No, it is not continuous}\end{array} $

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Toolbox:
  • If $f$ is a real function on a subset of the real numbers and $c$ a point in the domain of $f$, then $f$ is continous at $c$ if $\lim\limits_{x\to c} f(x) = f(c)$.
  • Sum, difference, product and quotient of any continuous functions is continous also.
Step 1:
$f(x)=\large\frac{1}{x-5}$
At $x=5$
$f(x)=\large\frac{1}{5-5}=\frac{1}{0}$
$\quad\quad\quad\quad\;\;\;=$Not defined.
Step 2:
When $x\neq 5\lim\limits_{\large x\to c}f(x)=\lim\limits_{\large x\to c}\large\frac{1}{x-5}=\frac{1}{c-5}$
$f(c)=\large\frac{1}{c-5}$
$f$ is continuous at $x\in R-\{5\}$
answered May 27, 2013 by sreemathi.v
 

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