Finding a simpler “local” lower bound for a rational function

I have obtained as the expression for some quantity the following gargantuan formula:

$$\frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$.

What I really need is a (very) good lower bound on it, that will hopefully be a more manageable expression.

Is there a systematic way of finding such bounds?

The bound needs to be valid only on $[5,\infty]$.

• $k$ is in $\mathbb{R}$? – Joseph O'Rourke Oct 24 '13 at 11:29
• this expression ranges from $-\infty$ to $\infty$ (for real $k$, and also for real positive $k$), so there is no lower bound. – Carlo Beenakker Oct 24 '13 at 11:31
• @JosephO'Rourke Yes. – Felix Goldberg Oct 24 '13 at 11:43
• @CarloBeenakker Sure, that's why I specified in my P.S. that I need actually a "local bound". I'll edit to make this point more prominent. Thanks. – Felix Goldberg Oct 24 '13 at 11:44
• As for a systematic way, your question reminds me of the problem of finding surrogate functions. A surrogate function is a function approximating the original one but cheaper to compute. – Waldemar Oct 24 '13 at 12:22

$$k^2+1.98-2.8/k$$

lower bound, error $<0.02$ for all $k>5$

• This looks great, but now I am curious - how did you obtain this result? – Felix Goldberg Oct 24 '13 at 13:42
• from the large-$k$ expansion – Carlo Beenakker Oct 24 '13 at 13:46
• Do you mean the Taylor series? Or something else? – Felix Goldberg Oct 24 '13 at 13:49
• Taylor series around $1/k=0$ (it works because 1/5 is small enough respect to unity) – Carlo Beenakker Oct 24 '13 at 14:40

The function blows up at approximately $-2.6$ and $+0.5$ and $+0.7$, and otherwise looks something like a parabola. It is unclear what is meant by a "lower bound" for such a function.  • Suppose we zoom in on $k\geq 5$ - can you suggest an adequate bound in that case? – Felix Goldberg Oct 24 '13 at 11:45
• @FelixGoldberg: Well, then the lower bound is just the value of the expression at $k=5$, which is $\frac{373393}{14125} \approx 26.4$. – Joseph O'Rourke Oct 24 '13 at 11:55
• I want a bound which is also a function of $k$... – Felix Goldberg Oct 24 '13 at 12:07
• @FelixGoldberg: I'm sorry, Felix, we are talking past one another. Your function is monotonically increasing $k \ge 5$. I am not sure what it means to express this as a function of $k$. – Joseph O'Rourke Oct 24 '13 at 12:33
• Now I see, from Waldemar's comment and Carlo's answer. – Joseph O'Rourke Oct 24 '13 at 13:16