I need some help. This can be a dumb question as I am not an electrical engineer nor an electronics engineer, but need to solve a problem as given below. I tried to solve it but not sure if it is true because spice simulations give different result.
Where is my mistake?
Problem
Choose Emitter Resistance \$R_E\$ for the given problem for the max Emitter Current \$I_E\$ that is possible. Take \$\beta=100\$.
My solution approach
\$I_C=\frac{V_+-V_{out}}{R_c}\Rightarrow V_c=R_cI_c+V_0\\ I_E=I_B+I_C=(\beta+1)I_B\\ \\ \frac{V_{out}-V_{CE}+V_+}{R_E}=I_E\\ \\ \frac{R_CI_C+V_+-V_{CE}+V_+}{R_E}=I_E\\ \\ I_C=\left( \frac{\beta}{1+\beta}\right)I_E\\ \\ \frac{R_C\left( \frac{\beta}{1+\beta}\right)I_E+V_+-V_{CE}+V_+}{R_E}=I_E\\ \$
which gives
\$ I_E=\cfrac{2V_+ -V_{CE}}{R_E-R_C \cfrac{\beta}{\beta+1}} \$
for the maximum current \${R_E-R_C \cfrac{\beta}{\beta+1}}\$ should be zero, thus,
\${R_E=R_C \cfrac{\beta}{\beta+1}}\$