Question

How do you find $\lim \dfrac{{\cos x}}{x}$ as $x \to 0$?

Answer 1

To find the limit of the given function $\dfrac{{\cos \left( x \right)}}{x}$, we need to just substitute the given value of $x$, i.e., as $x \to 0$, we can find the limit of the given function with respect to the value of $x$ it tends to as the limit of a constant function is the constant.

Complete step by step answer:
Let us write the given data:
$\mathop {\lim }\limits_{x \to 0} \dfrac{{\cos \left( x \right)}}{x}$
As, $x \to 0$, $\cos x \to \cos 0$; and also, we know that,
$\cos 0 = 1$ and $x \to 0$, hence we get:
$\Rightarrow \mathop {\lim }\limits_{x \to 0} \dfrac{{\cos \left( x \right)}}{x} = \dfrac{1}{0}$
$\Rightarrow \mathop {\lim }\limits_{x \to 0} \dfrac{{\cos \left( x \right)}}{x} = \infty$

Additional information: Here are some of the properties to find the limit functions:
Sum Rule: This rule states that the limit of the sum of two functions is equal to the sum of their limits.
Constant Function Rule: The limit of a constant function is the constant.
Constant Multiple Rule: The limit of a constant times a function is equal to the product of the constant and the limit of the function.
Product Rule: This rule says that the limit of the product of two functions is the product of their limits.
Quotient Rule: The limit of quotients of two functions is the quotient of their limits, provided that the limit in the denominator function is not zero.

Note: For a limit approaching the given value, the original functions must be differentiable on either side of the value, but not necessarily at the value given. The limit of a quotient is equal to the quotient of the limits. The limit of a constant function is equal to the constant. The limit of a linear function is equal to the number $x$ is approaching.