Question

$lim_{x
ightarrow0}\frac{2x^{6}+6x^{3}}{4x^{5}+3x^{3}}$ is

Explanation:

Step1: Factor out the lowest - power of x

Factor out $x^{3}$ from the numerator and denominator. The numerator $2x^{6}+6x^{3}=x^{3}(2x^{3} + 6)$ and the denominator $4x^{5}+3x^{3}=x^{3}(4x^{2}+3)$.
So, $\lim_{x
ightarrow0}\frac{2x^{6}+6x^{3}}{4x^{5}+3x^{3}}=\lim_{x
ightarrow0}\frac{x^{3}(2x^{3}+6)}{x^{3}(4x^{2}+3)}$.

Step2: Cancel out the common factor

Cancel out the common factor $x^{3}$ (since $x
eq0$ when taking the limit). We get $\lim_{x
ightarrow0}\frac{2x^{3}+6}{4x^{2}+3}$.

Step3: Substitute x = 0

Substitute $x = 0$ into $\frac{2x^{3}+6}{4x^{2}+3}$. When $x = 0$, we have $\frac{2(0)^{3}+6}{4(0)^{2}+3}=\frac{6}{3}=2$.

Answer:

$2$