- Started with
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
49.9
- Using strategy
rm 49.9
- Applied flip-- to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
50.1
- Applied associate-*l/ to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}}{2}\]
50.0
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
50.0
- Applied flip-+ to get
\[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}}} \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
49.8
- Applied frac-times to get
\[\frac{\color{red}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}} \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2} \leadsto \frac{\color{blue}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1}{\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}{2}\]
49.8
- Applied frac-sub to get
\[\frac{\color{red}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1}{\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{\frac{1}{\varepsilon} + 1}}}{2} \leadsto \frac{\color{blue}{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{\left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)}}}{2}\]
50.3
- Applied simplify to get
\[\frac{\frac{\color{red}{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot 1\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{\left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\frac{\color{blue}{\left(1 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 - 1\right)\right) \cdot \left(e^{x \cdot \left(1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right)}}{\left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2}\]
50.3
- Applied taylor to get
\[\frac{\frac{\left(1 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) - \left(e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 - 1\right)\right) \cdot \left(e^{x \cdot \left(1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right)}{\left(\left(1 - \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\frac{1}{\varepsilon} + 1\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
0.1
- Taylor expanded around 0 to get
\[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
0.1
