Question

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

Answer 1

Hint: 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.