Question

How do you find the limit of $\dfrac{{\sin x}}{x}$ as $x$ approaches to $\infty$?

Answer 1

To find the limit of the given function $\dfrac{{\sin x}}{x}$, apply squeeze theorem. As it deals with the limit values, rather than function values and if we have some function f(x), we define new functions h(x), g(x) such that $h\left( x \right) \leqslant f\left( x \right) \leqslant g\left( x \right)$.

Complete step by step solution:
Let us write the given function as
$\dfrac{{\sin x}}{x}$
We'll use the Squeeze Theorem, which states that if we have some function f(x), we define new functions h(x), g(x) such that
$h\left( x \right) \leqslant f\left( x \right) \leqslant g\left( x \right)$
Then, we'll take
$\mathop {\lim }\limits_{x \to a} h\left( x \right)$, $\mathop {\lim }\limits_{x \to a} g\left( x \right)$
If these limits are equal, then
$\mathop {\lim }\limits_{x \to a} f\left( x \right) = \mathop {\lim }\limits_{x \to a} h\left( x \right) = \mathop {\lim }\limits_{x \to a} g\left( x \right)$
Hence, all the limits are equal.
We know that
$- 1 \leqslant \sin x \leqslant 1$
Then,
$- \dfrac{1}{x} \leqslant \dfrac{{\sin x}}{x} \leqslant \dfrac{1}{x}$
Take
$\mathop {\lim }\limits_{x \to \infty } - \dfrac{1}{x},$
As $x \to \infty$, we get
$\mathop {\lim }\limits_{x \to \infty } - \dfrac{1}{x} = - \dfrac{1}{\infty } = 0$
$\mathop {\lim }\limits_{x \to \infty } \dfrac{1}{x} = \dfrac{1}{\infty } = 0$

Hence,
$\mathop {\lim }\limits_{x \to \infty } \dfrac{{\sin x}}{x} = 0$.

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 (if they exist).
Quotient Rule: The limit of quotient 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 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.