Show that f(x)=1-x+5x^2

Question

$6$. Let $$f(x) =\frac{9x^2 + 4}{(2x + 1)(x − 2)^2}$$ (i) Express $f(x)$ in partial fractions.

(ii) Show that, when $x$ is sufficiently small for $x^3$ and higher powers to be neglected, $$f(x) = 1 − x + 5x^2.$$

The answer for the first part is $f(x)= \frac{1}{2x+1} + \frac{4}{x-2} + \frac{8}{(x-2)^2}$ I still can't figure out the second part though. Please give me a step by step guide on how to answer this question.

Thanks.

Answer 1

Expand each term as a Taylor series. Throw away anything in $x^3$ or above. Hopefully, what is left is the expected result.