<s>[INST] Functions have similar graphs I graphed the two functions $y=x^2\sin\left(\frac{x}{50}\right)$ and $y=\frac{1}{50}x^3$ and I noticed they have extremely similar graphs from $-50 < x <50$. What is the reason for this?
 [/INST] Hint. By differentiation, one may prove that
$$
u-\frac{u^3}6\le\sin u\le u,\qquad u \in [0,1]
$$ giving
$$
\frac{x^3}{50}-\frac{x^5}{750000}\le x^2 \sin \left(\frac{x}{50}\right)\le \frac{x^3}{50}, \quad -1<\frac{x}{50}<1
$$ or

$$
\left| x^2 \sin \left(\frac{x}{50}\right)-\frac{x^3}{50}\right|\le \left|\frac{x^5}{750000}\right|, \quad -1<\frac{x}{50}<1.
$$

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