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Given $\lim \limits_{n\to \infty}\frac{a_n}{b_n}=1$

is this claim $\lim \limits_{n\to \infty}(a_n-b_n)=0$ false or true?

I think it's false because I can't talk about $\lim \limits_{n\to \infty}a_n$ nor about $\lim \limits_{n\to \infty}b_n$ however, I can't find an example that will disprove the claim.

Thank you.

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up vote 5 down vote accepted

Hint: try a quotient of polynomials of the same degree.

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