Mathematics

This limit tests whether you can distinguish between functions that look similar near zero. Both $\ln(1+x)$ and $\sin x$ approach 0 as $x \to 0$, so you'll get the indeterminate form $\frac{0}{0}$—the key is to expand both using Taylor series and see which terms survive after cancellation. Once you subtract them in the numerator, you'll find the lowest-order term that doesn't vanish, then divide by $x^3$ to find the limit.

This limit tests whether you can distinguish between functions that look similar near zero. Both $\ln(1+x)$ and $\sin x$ approach 0 as $x \to 0$, so you'll get the indeterminate form $\frac{0}{0}$—the key is to expand both using Taylor series and see which terms survive after cancellation. Once you subtract them in the numerator, you'll find the lowest-order term that doesn't vanish, then divide by $x^3$ to find the limit.