Question

Evaluate the limit:
$\mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{x - \sin x}}{x}} \right)\sin \left( {\dfrac{1}{x}} \right)$
A) 1
B) $ - 1$
C) Does not exist
D) 0

Answer 1

Hint:
Here, we will first try to simplify the first fraction by splitting its denominator. Then by using the formulas of the relation between trigonometry and limits, we will be able to simplify it further and find the required answer to this limit.

Complete step by step solution:
Given limit to solve: $\mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{x - \sin x}}{x}} \right)\sin \left( {\dfrac{1}{x}} \right)$
Let $f\left( x \right) = \left( {\dfrac{{x - \sin x}}{x}} \right)$.
Rewriting the equation, we get
$ \Rightarrow f\left( x \right) = \left( {\dfrac{x}{x} - \dfrac{{\sin x}}{x}} \right)$
Simplifying the expression, we get
$ \Rightarrow f\left( x \right) = \left( {1 - \dfrac{{\sin x}}{x}} \right)$
Now, we will evaluate the limit.
$\mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{x - \sin x}}{x}} \right) = \mathop {\lim }\limits_{x \to 0} \left( {1 - \dfrac{{\sin x}}{x}} \right)$
Splitting the limit function through subtraction, we get
$ \Rightarrow \mathop {\lim }\limits_{x \to 0} \left( {1 - \dfrac{{\sin x}}{x}} \right) = \mathop {\lim }\limits_{x \to 0} \left( 1 \right) - \mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{\sin x}}{x}} \right)$
Now, we know that, $\mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{\sin x}}{x}} \right) = 1$
Hence, substituting this value and removing the limits, we get,
$ \Rightarrow \mathop {\lim }\limits_{x \to 0} \left( {1 - \dfrac{{\sin x}}{x}} \right) = 1 - 1 = 0$
Therefore, $\mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{x - \sin x}}{x}} \right)\sin \left( {\dfrac{1}{x}} \right) = 0 \times \mathop {\lim }\limits_{x \to 0} \sin \left( {\dfrac{1}{x}} \right)$
Now, anything multiplied by 0, will always give the answer as 0.
Thus, the given limit $\mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{x - \sin x}}{x}} \right)\sin \left( {\dfrac{1}{x}} \right) = 0$

Hence, option D is the correct answer.

Note:
In mathematics, a limit is a value that a function approaches as the input or the index approaches some value. Limits are an essential element of calculus and are used to define continuity, integrals and derivatives. The idea of limits was first developed by ‘Archimedes of Syracuse’ to measure curved figures and the volume of a sphere in the third century B.C.