Answer
$\lim_{x\to0}\Big(\dfrac{-5x^{20}-2x^{2}+3000}{x^{2}-1}\Big)^{1/3}=-10\sqrt[3]{3}$
Work Step by Step
$\lim_{x\to0}\Big(\dfrac{-5x^{20}-2x^{2}+3000}{x^{2}-1}\Big)^{1/3}$
Apply the Limit of a Power law:
$\lim_{x\to0}\Big(\dfrac{-5x^{20}-2x^{2}+3000}{x^{2}-1}\Big)^{1/3}=...$
$...=\Big[\lim_{x\to0}\Big(\dfrac{-5x^{20}-2x^{2}+3000}{x^{2}-1}\Big)\Big]^{1/3}=...$
Apply the Limit of a Quotient law:
$...=\Big[\dfrac{\lim_{x\to0}(-5x^{20}-2x^{2}+3000)}{\lim_{x\to0}(x^{2}-1)}\Big]^{1/3}=...$
Apply the Limit of a Sum and a Difference in the numerator and in the denominator:
$...=\Big[\dfrac{\lim_{x\to0}-5x^{20}-\lim_{x\to0}2x^{2}+\lim_{x\to0}3000}{\lim_{x\to0}x^{2}-\lim_{x\to0}1}\Big]^{1/3}=...$
Apply the Limit of a Constant Multiple law to the first two terms in the numerator and the Limit of a Power law to the first term of the denominator:
$...=\Big[\dfrac{-5\lim_{x\to0}x^{20}-2\lim_{x\to0}x^{2}+\lim_{x\to0}3000}{(\lim_{x\to0}x)^{2}-\lim_{x\to0}1}\Big]^{1/3}=...$
Apply the Limit of a Power law to the first two terms of the numerator:
$...=\Big[\dfrac{-5(\lim_{x\to0}x)^{20}-2(\lim_{x\to0}x)^{2}+\lim_{x\to0}3000}{(\lim_{x\to0}x)^{2}-\lim_{x\to0}1}\Big]^{1/3}=...$
Evaluate:
$...=\Big[\dfrac{-5(0)^{20}-2(0)^{2}+3000}{(0)^{2}-1}\Big]^{1/3}=\Big(\dfrac{3000}{-1}\Big)^{1/3}=...$
$...=\sqrt[3]{-3000}=-10\sqrt[3]{3}$
