$lim_{x\rightarrow0}\frac{2x^{6}+6x^{3}}{4x^{5}+3x^{3}}$ is
$lim_{x\rightarrow0}\frac{2x^{6}+6x^{3}}{4x^{5}+3x^{3}}$ is
Answer
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, $\lim_{x\rightarrow0}\frac{2x^{6}+6x^{3}}{4x^{5}+3x^{3}}=\lim_{x\rightarrow0}\frac{x^{3}(2x^{3}+6)}{x^{3}(4x^{2}+3)}$.
Step2: Cancel out the common factor
Cancel out the common factor $x^{3}$ (since $x\neq0$ when taking the limit). We get $\lim_{x\rightarrow0}\frac{2x^{3}+6}{4x^{2}+3}$.
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:
$2$