Question

45-60: Find the limit.

52. \(\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{{\tan 7\theta }}\)

Short answer

The required value is \(\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{{\tan 7\theta }} = \frac{1}{7}\).

Step-by-step solution

Special Trigonometric limits

The special trigonometric for sine function is given below:

\(\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{\theta } = 1\)

Limit evaluation

The required limit is \(\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{{\tan 7\theta }}\).

Manipulate and solve the given limit as follows:

\(\begin{aligned}\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{{\tan 7\theta }} &= \mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{{\frac{{\sin 7\theta }}{{\cos 7\theta }}}}\\ &= \mathop {\lim }\limits_{\theta \to 0} \frac{{\cos 7\theta \sin \theta }}{{\sin 7\theta }}\\ &= \mathop {\lim }\limits_{\theta \to 0} \sin \theta \cdot \frac{{\cos 7\theta }}{{\sin 7\theta }} \cdot \frac{{7\theta }}{{7\theta }}\\ &= \mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{\theta } \cdot \frac{{7\theta }}{{\sin 7\theta }} \cdot \frac{{\cos 7\theta }}{7}\\ &= \mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{\theta } \cdot \mathop {\lim }\limits_{\theta \to 0} \frac{1}{{\frac{{\sin 7\theta }}{{7\theta }}}} \cdot \mathop {\lim }\limits_{\theta \to 0} \frac{{\cos 7\theta }}{7}\end{aligned}\)

Solve the above limit further as follows:

\(\begin{aligned}\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{{\tan 7\theta }} &= \mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{\theta } \cdot \mathop {\lim }\limits_{7\theta \to 0} \frac{1}{{\frac{{\sin 7\theta }}{{7\theta }}}} \cdot \mathop {\lim }\limits_{\theta \to 0} \frac{{\cos 7\theta }}{7}\\ &= 1 \cdot 1 \cdot \mathop {\lim }\limits_{\theta \to 0} \frac{{\cos 7\theta }}{7}\\ &= \frac{{\cos 0}}{7}\\ &= \frac{1}{7}\end{aligned}\)

Hence, \(\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{{\tan 7\theta }} = \frac{1}{7}\).