Question

  lim x + 2x² − x³/ 3 − x²

Answer 1

$$\lim _{x \rightarrow \infty}\left(\frac{x+2 x^{2}-x^{3}}{3-x^{2}}\right)\\ \text{Divide by highest denominator power: } \frac{\frac{1}{x}+2-x}{\frac{3}{x^{2}}-1}\\[2mm] \begin{aligned} &=\lim _{x \rightarrow \infty}\left(\frac{\frac{1}{x}+2-x}{\frac{3}{x^{2}}-1}\right) \\ &\lim _{x \rightarrow a}\left[\frac{f(x)}{g(x)}\right]=\frac{\lim _{x \rightarrow a f} f(x)}{\lim _{x \rightarrow a g} g(x)}, \quad \lim _{x \rightarrow a g}(x) \neq 0 \end{aligned} \\[2mm] \textit{With the exception of indeterminate form}\\[2mm] =\frac{\lim _{x \rightarrow \infty}\left(\frac{1}{x}+2-x\right)}{\lim _{x \rightarrow \infty}\left(\frac{3}{x^{2}}-1\right)}\\ \lim _{x \rightarrow \infty}\left(\frac{1}{x}+2-x\right)=-\infty\\ \lim _{x \rightarrow \infty}\left(\frac{3}{x^{2}}-1\right)=-1\\ =\frac{-\infty}{-1}\\ =\infty$$