Problem 819 (difficulty: 4/10)

Assume that \(\displaystyle f,g\) are \(\displaystyle k\) times differentiable, \(\displaystyle \lim\limits_\infty|g|=\infty\), \(\displaystyle g^{(k)}\ne0\) and \(\displaystyle \lim\limits_\infty\frac{f^{(k)}}{g^{(k)}}=\beta\). Does it imply that \(\displaystyle \lim\limits_\infty\frac{f}{g}=\beta\)?

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