find $lim_{x\rightarrowinfty}\frac{3x^{3}-5x}{x^{3}-2x^{2}+1}$. choose 1 answer: a -5 b 0 c 3 d the limit is…

Explanation:

Step1: Divide by highest - power of x

Divide both the numerator and denominator by $x^{3}$ since the highest - power of $x$ in the denominator is $x^{3}$. [ \begin{align*} \lim_{x\rightarrow\infty}\frac{3x^{3}-5x}{x^{3}-2x^{2}+1}&=\lim_{x\rightarrow\infty}\frac{\frac{3x^{3}}{x^{3}}-\frac{5x}{x^{3}}}{\frac{x^{3}}{x^{3}}-\frac{2x^{2}}{x^{3}}+\frac{1}{x^{3}}}\ &=\lim_{x\rightarrow\infty}\frac{3-\frac{5}{x^{2}}}{1 - \frac{2}{x}+\frac{1}{x^{3}}} \end{align*} ]

Step2: Evaluate limits of individual terms

As $x\rightarrow\infty$, $\lim_{x\rightarrow\infty}\frac{1}{x}=0$, $\lim_{x\rightarrow\infty}\frac{1}{x^{2}} = 0$ and $\lim_{x\rightarrow\infty}\frac{1}{x^{3}}=0$. [ \begin{align*} \lim_{x\rightarrow\infty}\frac{3-\frac{5}{x^{2}}}{1 - \frac{2}{x}+\frac{1}{x^{3}}}&=\frac{\lim_{x\rightarrow\infty}(3)-\lim_{x\rightarrow\infty}\frac{5}{x^{2}}}{\lim_{x\rightarrow\infty}(1)-\lim_{x\rightarrow\infty}\frac{2}{x}+\lim_{x\rightarrow\infty}\frac{1}{x^{3}}}\ &=\frac{3 - 0}{1-0 + 0}\ &=3 \end{align*} ]

Answer:

C. 3