Discussion
The value of \(\lim_{x \to 0} \frac{x^{3} - \sin(x)}{x}\) is
Answer: D
The limit is -1.
Split the fraction:
\[ \begin{aligned} \frac{x^{3} - \sin x}{x} &= x^{2} - \frac{\sin x}{x} \\ &\to 0 - 1 = -1 \end{aligned} \]
using the standard limit \(\sin x / x \to 1\). L'Hospital's rule gives the same: \((3x^{2} - \cos x)/1 \to -1\). 1 drops the minus sign, 0 treats the whole numerator as negligible, and 3 comes from differentiating \(x^{3}\) but forgetting the sine term.
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