User:HankskyjamesMCPE/sandbox/From other users

Example of a parametric function from two parameter functions: $$ \left(x_t(t),y_t(t)\right)= \begin{cases} x_t(t)&=t^5-4t \\ y_t(t)&=t^4-t \end{cases} $$ The derivative of a parametric function is defined by: $$ m(t)\overset{\underset{\mathrm{def}}{}}{=}\frac{\dot{y}_t(t)}{\dot{x}_t(t)}\implies m(t)=\frac{4t^3-1}{5t^4-4} $$ Such that the tangent equation at a specific time $$t=t_1$$ becomes: $$ y-y_t(t_1)=m(t_1)\left(x-x_t(t_1)\right)\implies y=\frac{4t_1^3-1}{5t_1^4-4}\left(x-t_1^5+4t_1\right)+t_1^4-t_1 $$ $$\mbox{if }~\sqrt[n_1]{a_1}\pm\sqrt[n_2]{a_2}\pm\sqrt[n_3]{a_3}\pm\ldots=0\mbox{ and }n_1,n_2,n_3,\ldots\geq1,~n\in\mathbb{Z}\mbox{ then}$$ $$\lim_{x\to\infty}\left(\sqrt[n_1]{a_1x^{n_1}+b_1x^{n_1-1}+c_1x^{n_1-2}+\ldots}\pm\sqrt[n_2]{a_2x^{n_2}+b_2x^{n_2-1}+c_2x^{n_2-2}+\ldots}\pm\ldots\right)=\frac{b_1}{n_1\sqrt[n_1]{a_1^{n_1-1}}}\pm\frac{b_2}{n_2\sqrt[n_2]{a_2^{n_2-1}}}\pm\ldots $$