Question

45-60: Find the limit.

56. \(\mathop {\lim }\limits_{x \to 0} \csc x \cdot \sin \left( {\sin x} \right)\)

Short answer

The required value is \(\mathop {\lim }\limits_{x \to 0} \csc x \cdot \sin \left( {\sin x} \right) = 1\).

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 \(\mathop {\lim }\limits_{x \to 0} \csc x \cdot \sin \left( {\sin x} \right)\).

Manipulate and solve the given limit as follows:

\(\begin{aligned}\mathop {\lim }\limits_{x \to 0} \csc x \cdot \sin \left( {\sin x} \right) &= \mathop {\lim }\limits_{x \to 0} \frac{1}{{\sin x}} \cdot \sin \left( {\sin x} \right)\\ &= \mathop {\lim }\limits_{x \to 0} \frac{{\sin \left( {\sin x} \right)}}{{\sin x}}\\ &= \mathop {\lim }\limits_{\sin x \to 0} \frac{{\sin \left( {\sin x} \right)}}{{\sin x}}\\ &= 1\end{aligned}\)

Hence, \(\mathop {\lim }\limits_{x \to 0} \csc x \cdot \sin \left( {\sin x} \right) = 1\).