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If $f(x)=2 x$ and $g(x)=\frac{x^2}{2}+1$, then which of the following can be a discontinuous function?
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Verified Answer
The correct answer is:
$\frac{g(x)}{f(x)}$
$\frac{g(x)}{f(x)}$
We know that, sum, product and difference of two polynomials is a polynomials, and polynomial function is everywhere continuous.
Now, we check the continuity of $\frac{g(x)}{f(x)}$
$$
\frac{g(x)}{f(x)}=\frac{\frac{x^2}{2}+1}{2 x}
$$
Clearly, $\frac{\mathrm{g}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}$ is not defined at $\mathrm{x}=0$
$\therefore$ It is discontinuous at $\mathrm{x}=0$
Now, we check the continuity of $\frac{g(x)}{f(x)}$
$$
\frac{g(x)}{f(x)}=\frac{\frac{x^2}{2}+1}{2 x}
$$
Clearly, $\frac{\mathrm{g}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}$ is not defined at $\mathrm{x}=0$
$\therefore$ It is discontinuous at $\mathrm{x}=0$
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