Question

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

Answer 1

Hint : Here in this question we need to find the limit for the function. Let we define the given function as $f(x)$ and we apply the limit to it. The function will get closer and closer to some number. When the x approaches to infinity we need to find the limit of a function.

Complete step-by-step answer :
The idea of a limit is a basis of all calculus. The limit of a function is defined as let $f(x)$be a function defined on an interval that contains$x = a$. Then we say that$\mathop {\lim }\limits_{x \to a} f(x) = L$, if for every $\varepsilon > 0$there is some number $\delta > 0$such that $\left| {f(x) - L} \right| < \varepsilon$whenever$0 < \left| {x - a} \right| < \delta$.
Here we have to find the value of the limit when x is infinity. The function is trigonometry. Let we define the given function as$f(x) = \dfrac{{\sin x}}{x}$. Now we are going to apply the limit to the function $f(x)$so we have
$\mathop {\lim }\limits_{x \to \infty } f(x) = \mathop {\lim }\limits_{x \to \infty } \dfrac{{\sin x}}{x}$
The function $f(x)$ is a trigonometric function. As x approaches $\infty$ we have to substitute the x as$\infty$.
When we consider the x value we will consider it as infinity. So by applying limit we have
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } f(x) = \dfrac{{\sin (\infty )}}{\infty }$
The sine function of infinity is a finite value and the any number divided by infinity is zero so we have
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } f(x) = 0$
Hence the when the x approaches to zero for the function $\dfrac{{\sin x}}{x}$
Therefore we have found the limit for the function $f(x) = \dfrac{{\sin x}}{x}$
Hence $\mathop {\lim }\limits_{x \to \infty } \dfrac{{\sin x}}{x} = 0$
So, the correct answer is “0”.

Note : The function is of trigonometry and it is in form of fraction. The sine function of infinity is some finite value. When the x approaches to infinity, it implies that function is going to the infinity and the value will be zero. Infinity is a large number and doesn't know the exact value. Any number divided by infinity will be always zero.