Question

1. $lim_{x

ightarrowinfty}\frac{3x^{2}+x^{2}-x - 1}{x^{3}+5x + 2}$

Explanation:

Step1: Divide numerator and denominator by highest - power of z

Divide both the numerator $3z^{2}+z^{2}-z - 1=4z^{2}-z - 1$ and the denominator $z^{2}+5z + 2$ by $z^{2}$. We get $\lim_{z
ightarrow\infty}\frac{4z^{2}/z^{2}-z/z^{2}-1/z^{2}}{z^{2}/z^{2}+5z/z^{2}+2/z^{2}}=\lim_{z
ightarrow\infty}\frac{4 - \frac{1}{z}-\frac{1}{z^{2}}}{1+\frac{5}{z}+\frac{2}{z^{2}}}$.

Step2: Apply limit rules

As $z
ightarrow\infty$, we know that $\lim_{z
ightarrow\infty}\frac{1}{z}=0$ and $\lim_{z
ightarrow\infty}\frac{1}{z^{2}} = 0$. So, $\lim_{z
ightarrow\infty}\frac{4 - \frac{1}{z}-\frac{1}{z^{2}}}{1+\frac{5}{z}+\frac{2}{z^{2}}}=\frac{4-0 - 0}{1+0+0}$.

Answer:

4