Question

How do you evaluate the limit of $\dfrac{x}{{\sin \left( x \right)}}$ as x approaches zero ?

Answer 1

Hint : The given question requires us to evaluate a limit. The limit given to us is of indeterminate form and hence can be solved easily using the L’Hospital’s rule. There are various methods and steps to evaluate a limit. Some of the common steps while solving limits involve rationalization and applying some basic results on frequently used limits. L’Hospital’s rule involves solving limits of indeterminate form by differentiating both numerator and denominator with respect to the variable separately and then applying the required limit.

Complete step by step solution:
We have to evaluate the limit $\lim \left( {\dfrac{x}{{\sin \left( x \right)}}} \right)$ as $x \to 0$ using L’Hospital’s rule.
So, if we put the limit x tending to zero into the expression $\dfrac{x}{{\sin \left( x \right)}}$ , we get an indeterminate form limit. Hence, L’Hospital’s rule can be applied here to find the value of the concerned limit.
So, Applying L’Hospital’s rule, we have to differentiate both numerator and denominator with respect to the variable x separately and then apply the limit.
Hence, $\mathop {\lim }\limits_{x \to 0} \left( {\dfrac{x}{{\sin \left( x \right)}}} \right)$
Now, the derivative of $\sin \left( x \right)$ can be evaluated as $\cos \left( x \right)$ using the differentiation of trigonometric functions and the derivative of x is $1$ using the power rule of differentiation.
$= \mathop {\lim }\limits_{x \to 0} \left( {\dfrac{1}{{\cos \left( x \right)}}} \right)$
Now, substituting in the limit of the variable. We get,
$= \left( {\dfrac{1}{{\cos \left( 0 \right)}}} \right)$
$= \left( {\dfrac{1}{1}} \right) = 1$
So, the value of the limit $\lim \left( {\dfrac{x}{{\sin \left( x \right)}}} \right)$ as $x \to 0$ is $1$ .
So, the correct answer is “1”.

Note : The given question tests our concepts of limits and differentiation. One must know about the L’Hopital’s rule of evaluating the limits and its applications in order to attempt such questions. Care should be taken while simplifying the expressions and carrying out calculations.