Question

How do you find the limit $\lim \left( {\dfrac{{1 - \cos x}}{x}} \right)$ as x approaches $0$ using L’Hospital’s rule.

Answer 1

Hint: The given question requires us to evaluate a limit. 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 $\lim \left( {\dfrac{{1 - \cos x}}{x}} \right)$ as x approaches $0$ using L’Hospital’s rule. So, if we put the limit x tending to zero into the expression $\left( {\dfrac{{1 - \cos x}}{x}} \right)$, 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, $\mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{1 - \cos x}}{x}} \right)$
Now, the derivative of $\left( {1 - \cos x} \right)$ can be evaluated as $\sin x$ using the addition rule of differentiation and the derivative of x is $1$ using the power rule of differentiation.
$ = \mathop {\lim }\limits_{x \to 0} \left( {\dfrac{{\sin x}}{1}} \right)$
$ = \left( {\dfrac{{\sin 0}}{1}} \right)$
$ = 0$

So, the value of the limit $\lim \left( {\dfrac{{1 - \cos x}}{x}} \right)$ as x approaches zero is $0$ .

Note: The given question can also be solved by various other methods. We must remember some basic limits to solve such questions involving complex limits. We can also manipulate the limits to resemble a standard form of limit and then calculate it accordingly.