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Let $f(x)=\left\{\begin{array}{ll}3 x-4, & 0 \leq x \leq 2 \\ 2 x+\ell, & 2 < x \leq 9\end{array}\right.$
If $\mathrm{f}$ is continuous at $\mathrm{x}=2$, then what is the value of $\ell$ ?
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If $\mathrm{f}$ is continuous at $\mathrm{x}=2$, then what is the value of $\ell$ ?
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Verified Answer
The correct answer is:
-2
Given function is :
$f(x)=\left\{\begin{array}{ll}3 x-4, & 0 \leq x \leq 2 \\ 2 x+\ell, & 2 < x \leq 9\end{array}\right.$
and also given that $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=2$ For a function to be continuous at a point $\mathrm{LHL}=\mathrm{RHL}$ $=\mathrm{V} . \mathrm{F}$. at that point. $\mathrm{f}(2)=2=\mathrm{V}$.F.
$\Rightarrow \mathrm{RHL}: \lim _{\mathrm{x} \rightarrow 2}(2 \mathrm{x}+\ell)=3(2)-4$
$\Rightarrow \lim _{\mathrm{h} \rightarrow 0}\{2(2+\mathrm{h})+\ell\}=6-4$
$\Rightarrow 4+\ell=2$
$\Rightarrow \ell=-2$
$f(x)=\left\{\begin{array}{ll}3 x-4, & 0 \leq x \leq 2 \\ 2 x+\ell, & 2 < x \leq 9\end{array}\right.$
and also given that $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=2$ For a function to be continuous at a point $\mathrm{LHL}=\mathrm{RHL}$ $=\mathrm{V} . \mathrm{F}$. at that point. $\mathrm{f}(2)=2=\mathrm{V}$.F.
$\Rightarrow \mathrm{RHL}: \lim _{\mathrm{x} \rightarrow 2}(2 \mathrm{x}+\ell)=3(2)-4$
$\Rightarrow \lim _{\mathrm{h} \rightarrow 0}\{2(2+\mathrm{h})+\ell\}=6-4$
$\Rightarrow 4+\ell=2$
$\Rightarrow \ell=-2$
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