The CSAT and Practice[+] are designed by the Climb Foundation to help candidates. We are advocates for more opportunity to shine and less opportunity to fail, and we strive to level the playing field.

The Computer Laboratory

Practice Paper 2 Question 6

What does \({\lim\limits_{x\to\infty} \frac{f(x)}{f(-x)}=-1}\) imply about a polynomial \(f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0\) with real coefficients? Prove your answer.

The above links are provided as is. They are not affiliated with the Climb Foundation unless otherwise specified.


  • Hint 1
    Try to factorize \(x^n.\)
  • Hint 2
    ... then take the limit.
  • Hint 3
    What is \(\lim_{x\to\infty}\frac{1}{x^k}\) for any positive integer \(k?\)


Factorizing \(x^n\) and knowing \(\lim_{x\to\infty}\frac{1}{x^k}=0\) for any positive integer \(k\), we have:\[\begin{align} \lim\limits_{x\to\infty} \frac{f(x)}{f(-x)} &= \lim\limits_{x\to\infty} \frac{x^n(a_n+a_{n-1}/x+a_{n-2}/x^2+\cdots+a_0/x^n)} {(-x)^n(a_n+a_{n-1}/(-x)+a_{n-2}/(-x)^2+\cdots+a_0/(-x)^n)} \\ &= \lim_{x\to\infty} \frac{x^n(a_n+0+0+\cdots+0)}{(-x)^n(a_n+0+0+\cdots+0)} \\ &= (-1)^n. \end{align} \] If this limit is \(-1,\) then \(n\) must be odd.

If you have queries or suggestions about the content on this page or the CSAT Practice Platform then you can write to us at[email protected]. Please do not write to this address regarding general admissions or course queries.