What are the asymptotes and removable discontinuities, if any, of $f (x) = \frac{2 x - e^{x}}{2 - e^{x}}$ $f(x)= (2x-e^x)/(2-e^x)$ ?

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What are the asymptotes and removable discontinuities, if any, of $f (x) = \frac{2 x - e^{x}}{2 - e^{x}}$ $f(x)= (2x-e^x)/(2-e^x)$ ?

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Answer 1 · 2018-06-09T16:47:23

asymptote at:

$x = ln 2 \approx 0.693$ $x=ln2 ~~0.693$

No removable discontinuities.

Explanation:

$f (x) = \frac{2 x - e^{x}}{2 - e^{x}}$ $f(x)= (2x-e^x)/(2-e^x)$

There are no factors that cancel in the numerator and denominator so there are no removable discontinuities (holes). To find the asymptote(s) find when the function is undefined, i.e. $\frac{2 x - e^{x}}{0}$ $(2x-e^x)/0$

$2 - e^{x} = 0$ $2-e^x=0$

$e^{x} = 2$ $e^x=2$

$ln (e^{x}) = ln 2$ $ln(e^x)=ln2$

$x = ln 2 \approx 0.693$ $x=ln2 ~~0.693$

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What are the asymptotes and removable discontinuities, if any, of $f (x) = \frac{2 x - e^{x}}{2 - e^{x}}$ $f(x)= (2x-e^x)/(2-e^x)$ ?

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Explanation:

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