Question

Lim \[x\csc x\] \[x \to 0\] how to get the answer?

Answer 1

Hint: To solve the given trigonometric function with the limits, we must know all the basic relations of trigonometric identities and their formulas. To evaluate the given trigonometric function, apply trigonometric identity formulas i.e., applying \[\csc x\] formula in terms of sin function, as we know that sec, csc and cot are derived from primary functions of sin, cos and tan, hence using these functions we can evaluate the given function.

Complete step-by-step solution:
The given function:
\[\mathop {\lim }\limits_{x \to 0} x\csc x\]
We know that, the reciprocal identity of \[\csc \theta \] is:
\[\csc \theta = \dfrac{1}{{\sin \theta }}\]
Hence, applying it to the given function we get
\[\mathop {\lim }\limits_{x \to 0} x\csc x\]
\[\mathop {\lim }\limits_{x \to 0} x \times \dfrac{1}{{\sin x}}\]
We get,
\[ \Rightarrow \] \[\mathop {\lim }\limits_{x \to 0} \dfrac{x}{{\sin x}}\]
\[ \Rightarrow \] \[\mathop {\lim }\limits_{x \to 0} \dfrac{1}{{\dfrac{{\sin x}}{x}}}\]
We know that: \[\mathop {\lim }\limits_{x \to 0} \dfrac{{\sin x}}{x} = 1\]
Hence, we get:
$\Rightarrow$ \[\dfrac{1}{1}\]
$\Rightarrow 1$

Therefore, \[\mathop {\lim }\limits_{x \to 0} x\csc x\] = 1

Additional Information:
The three basic functions in trigonometry are sine, cosine and tangent. Based on these three functions the other three functions that are cotangent, secant and cosecant are derived.
All the trigonometrical concepts are based on these functions. Hence, to understand trigonometry further we need to learn these functions and their respective formulas at first.
If θ is the angle in a right-angled triangle, then
Sin θ = \[\dfrac{{perpendicular}}{{hypotenuse}}\]
Cos θ = \[\dfrac{{base}}{{hypotenuse}}\]
Tan θ = \[\dfrac{{perpendicular}}{{base}}\]
The other three functions i.e., cot, sec and cosec depend on tan, cos and sin respectively.

Note: The key point to prove any trigonometric function is to note the formulas of all related functions with respect to the equation, and evaluate with respect to the given. And here are some of the formulas to be noted.
\[\csc \theta = \dfrac{1}{{\sin \theta }}\] , \[\cot \theta = \dfrac{1}{{\tan \theta }}\]
\[\sin \theta = \dfrac{1}{{\csc \theta }}\], \[\tan \theta = \dfrac{1}{{\cot \theta }}\]
\[\sec \theta = \dfrac{1}{{\cos \theta }}\],\[\cos \theta = \dfrac{1}{{\sec \theta }}\]