Question
Limit x-->0 tan mx/tan nx
Answers
GPT-4o mini
To determine the limit \(\lim_{x \to 0} \frac{\tan(mx)}{\tan(nx)}\), we can make use of the fact that for small values of \(x\), \(\tan(kx) \approx kx\). This approximation holds true because as \(x\) approaches 0, the tangent function behaves linearly.
1. Using the small-angle approximation:
\[
\tan(mx) \approx mx \quad \text{and} \quad \tan(nx) \approx nx
\]
2. Substitute these approximations into the limit:
\[
\frac{\tan(mx)}{\tan(nx)} \approx \frac{mx}{nx} = \frac{m}{n}
\]
3. Now, taking the limit as \(x\) approaches 0, we have:
\[
\lim_{x \to 0} \frac{\tan(mx)}{\tan(nx)} = \lim_{x \to 0} \frac{mx}{nx} = \frac{m}{n}
\]
Thus, the limit is:
\[
\boxed{\frac{m}{n}}
\]
1. Using the small-angle approximation:
\[
\tan(mx) \approx mx \quad \text{and} \quad \tan(nx) \approx nx
\]
2. Substitute these approximations into the limit:
\[
\frac{\tan(mx)}{\tan(nx)} \approx \frac{mx}{nx} = \frac{m}{n}
\]
3. Now, taking the limit as \(x\) approaches 0, we have:
\[
\lim_{x \to 0} \frac{\tan(mx)}{\tan(nx)} = \lim_{x \to 0} \frac{mx}{nx} = \frac{m}{n}
\]
Thus, the limit is:
\[
\boxed{\frac{m}{n}}
\]