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Find whether f(x) is continuous or discontinuous at the indicated point $f(x)=\left \{\begin{array}{1 1}x-4, & if\;x\neq 4\\2(x-4)\\0, & if\;x=4\end{array}\right.$at x=4.

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Toolbox:
  • A function is said to be discontinuous at $x=a$ if both the RHL and the LHL does not exist.
  • It is also discontinuous if RHL $\neq$ LHL.
Step 1:
Given : $f(x)=\left \{\begin{array}{1 1}x-4, & if\;x\neq 4\\2(x-4)\\0, & if\;x=4\end{array}\right.$at x=4.
Let $f(x)=x-4$
The LHL at $x=4^-$ is written as
$\lim\limits_{\large x\to 4^-}f(x)=\lim\limits_{\large x\to 4^-}(x-4)$
(i.e) $\lim\limits_{\large h\to 0}(4-h-4)=0$
Step 2:
Similarly the RHL at $x=4^+$ is written as
$\lim\limits_{\large x\to 4^+}f(x)=\lim\limits_{\large x\to 4^+}(x-4)$
(i.e) $\lim\limits_{\large h\to 0}(4+h-4)=0$
Step 3:
Similarly the LHL for $f(x)=2(x-4)$ at $x=4^-$ is written as
$\lim\limits_{\large x\to 4^-}f(x)=\lim\limits_{\large x\to 4^-}2(x-4)$
(i.e) $\lim\limits_{\large h\to 0}2(4-h)-4=4$
Step 4:
Similarly the RHL for $f(x)=2(x-4)$ at $x=4^+$ is written as
$\lim\limits_{\large x\to 4^+}f(x)=\lim\limits_{\large x\to 4^+}2(x-4)$
(i.e) $\lim\limits_{\large h\to 0}2(4+h)-4=4$
Hence it is continuous at all points when $x\neq 4$
Step 5:
When $x=4$
$f(x)=2(4-4)$
$\quad\;\;\;=0$
Hence it is not continuous at $x=4$
answered Jun 25, 2013 by sreemathi.v
 

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