Question

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

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 $x^{3}$.
\[

$$\begin{align*} \lim_{x ightarrow\infty}\frac{3x^{3}-5x}{x^{3}-2x^{2}+1}&=\lim_{x ightarrow\infty}\frac{\frac{3x^{3}}{x^{3}}-\frac{5x}{x^{3}}}{\frac{x^{3}}{x^{3}}-\frac{2x^{2}}{x^{3}}+\frac{1}{x^{3}}}\\ &=\lim_{x ightarrow\infty}\frac{3-\frac{5}{x^{2}}}{1 - \frac{2}{x}+\frac{1}{x^{3}}} \end{align*}$$

\]

Step2: Evaluate limits of individual terms

As $x
ightarrow\infty$, $\lim_{x
ightarrow\infty}\frac{1}{x}=0$, $\lim_{x
ightarrow\infty}\frac{1}{x^{2}} = 0$ and $\lim_{x
ightarrow\infty}\frac{1}{x^{3}}=0$.
\[

$$\begin{align*} \lim_{x ightarrow\infty}\frac{3-\frac{5}{x^{2}}}{1 - \frac{2}{x}+\frac{1}{x^{3}}}&=\frac{\lim_{x ightarrow\infty}(3)-\lim_{x ightarrow\infty}\frac{5}{x^{2}}}{\lim_{x ightarrow\infty}(1)-\lim_{x ightarrow\infty}\frac{2}{x}+\lim_{x ightarrow\infty}\frac{1}{x^{3}}}\\ &=\frac{3 - 0}{1-0 + 0}\\ &=3 \end{align*}$$

\]

Answer:

C. 3