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Examine the continuity of \(f\), where \(f\) is defined by $ f(x) = \left\{ \begin{array} {1 1} \sin x - \cos x,& \quad\text{ if \(x\) \(\neq\) 0 }\\ -1 ,& \quad \text{if $x$ =0}\\ \end{array} \right. $

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  • If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
LHL=$\lim\limits_{\large x\to 0}(\sin x-\cos x)$
$\quad\;\;=\lim\limits_{\large h\to 0}[\sin (-h)-\cos(-h)]$
$\quad\;\;=\lim\limits_{\large h\to 0}[-\sin h-\cos h]$
$\quad\;\;=-1$
LHL=-1
Step 2:
RHL=$\lim\limits_{\large x\to 0}(\sin x-\cos x)$
$\quad\;\;=\lim\limits_{\large h\to 0}[\sin h-\cos h]$
$\quad\;\;=-1$
RHL=-1
Step 3:
$f(0)=-1$
LHL=RHL=f(0)
$f$ is continuous at $x=0$
$\sin x$ and $\cos x$ both are continuous for all $x\in R$
Therefore $\sin x-\cos x$ is continuous for all $x\in R$
$\Rightarrow f$ is continuous for all $x\in R$.
answered May 29, 2013 by sreemathi.v
 

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