Use infinite series to evaluate $\lim_{x \rightarrow \infty} (x^3 – 5x^2 + 1)^{\frac{1}{3}} – x$

Question

Use infinite series to evaluate $\displaystyle\lim_{x \rightarrow \infty} (x^3 – 5x^2 + 1)^{\frac{1}{3}} – x$

I know the limit is $-\dfrac{5}{3}$ after rationalizing the expression, but I don't know how to prove it using Taylor series. Could someone give me any hints? I prefer hints to complete solutions.

Answer 1

Let $1/x=h$ to find

$$\lim_{h\to0^+}\dfrac{(1-5h+h^3)^{1/3}-1^{1/3}}h$$

Now rationalize the numerator using $$a^3-b^3=(a-b)(a^2+ab+b^2)$$