[Math] Evaluate limit as x approaches infinity of $\lim_{x\to\infty}\frac{\sqrt{x^3 +7x}}{\sqrt{4x^3+5}}$

Question

I am having trouble figuring out how to answer this question by determining the degree of the numerator and/or denominator:
$$\lim_{x\to\infty}\frac{\sqrt{x^3 +7x}}{\sqrt{4x^3+5}}$$
I have tried deriving the first coefficient of the numerator and denominator, but not sure how to proceed to find the limit as $x \to \infty$.

Answer 1

Notice, $$\lim_{x\to \infty}\frac{\sqrt{x^3+7x}}{\sqrt{4x^3+5}}$$ $$=\lim_{x\to \infty}\sqrt{\frac{x^3+7x}{4x^3+5}}$$ $$=\lim_{x\to \infty}\sqrt{\frac{x^3\left(1+\frac{7}{x^2}\right)}{x^3\left(4+\frac{5}{x^3}\right)}}$$ $$=\lim_{x\to \infty}\sqrt{\frac{1+\frac{7}{x^2}}{4+\frac{5}{x^3}}}$$ $$=\sqrt{\frac{1+0}{4+0}}=\color{red}{\frac{1}{2}}$$