The maximum value

Question:

The maximum value of $\mathrm{z}$ in the following equation $z=6 x y+y^{2}$, where $3 x+4 y \leq 100$ and $4 x+3 y \leq 75$ for $x \geq 0$ and $y \geq 0$ is___________.

Solution:

$\lim _{x \rightarrow 0} \frac{a e^{x}-b \cos x+c e^{-x}}{x \sin x}=2$

$\Rightarrow \lim _{x \rightarrow 0} \frac{a\left(1+x+\frac{x^{2}}{2 !} \cdots\right)^{-b}\left(1-\frac{x^{2}}{2 !}+\cdots\right)+c\left(1-x+\frac{x^{2}}{2 !}\right)}{\left(\frac{x \sin x}{x}\right) x}=2$

$a-b+c=0$

$a-c=0$

$\& \frac{a+b+c}{2}=2$

$\Rightarrow a+b+c=4$