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Find the values of $k$ so that the function $f$ is continuous at the indicated point in $f(x) = \left\{ \begin{array} {1 1} kx+1 ,& \quad\text{ if $ x $ \(\leq \pi\)}\\ \cos x,& \quad \text{if $x$ > \(\pi\)}\\ \end{array} \right.$ at $x = \pi$

$\begin{array}{1 1}k= \large\frac{3}{\pi} \\ k= \large\frac{-2}{\pi} \\ k= \large\frac{2}{\pi} \\ k= \large\frac{-1}{\pi} \end{array} $

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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:
At $x=\pi$
LHL=$\lim\limits_{\large x\to n}f(x)$
$\quad\;\;=\lim\limits_{\large x\to \pi}(kx+1)$
$\quad\;\;=k\pi+1$
Step 2:
RHL=$\lim\limits_{\large x\to \pi}\cos x=-1$
For continuity at $x=\pi$
$LHL=RHL=f(x)$
$k\pi+1=-1$
$k=\large\frac{-2}{\pi}$
Hence the value of $k$ is $\large\frac{-2}{\pi}$
answered May 29, 2013 by sreemathi.v
 

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