Evaluate the limit of sin(9x)/cot(12x) as x approaches 0.

Step-by-step solution

1. 1

   Recognise why direct substitution stalls. At $x=0$ the numerator $\sin(9x)$ tends to $0$, while $\cot(12x)=\cos(12x)/\sin(12x)$ blows up. A quotient of the form $0/\infty$ is not an indeterminate form, but the cotangent hides that fact, so rewrite it before deciding anything.

2. 2

   Replace the cotangent by sine and cosine. Using $\cot\theta=\dfrac{\cos\theta}{\sin\theta}$, dividing by $\cot(12x)$ is the same as multiplying by its reciprocal $\tan(12x)$: $\frac{\sin(9x)}{\cot(12x)}=\sin(9x)\cdot\frac{\sin(12x)}{\cos(12x)}=\frac{\sin(9x)\sin(12x)}{\cos(12x)}.$

3. 3

   Check what each factor does as $x\to 0$. Sine is continuous with $\sin 0=0$, so $\sin(9x)\to 0$ and $\sin(12x)\to 0$. Cosine is continuous with $\cos 0=1$, so $\cos(12x)\to 1$. Crucially the denominator no longer approaches $0$, so no cancellation or L'Hopital step is needed.

4. 4

   Substitute the limits. $\lim_{x\to 0}\frac{\sin(9x)\sin(12x)}{\cos(12x)}=\frac{0\cdot 0}{1}=0.$

5. 5

   Sanity-check numerically. At $x=0.01$ the expression equals $\sin(0.09)\tan(0.12)\approx 0.0899\times 0.1206\approx 0.01084$, and at $x=0.001$ it is about $0.000108$ — shrinking by roughly a factor of $100$ each time $x$ shrinks by $10$, exactly what the leading behaviour $108x^2$ predicts. The limit is $0$.