Question

How do I find the limit as x approaches infinity of the square root function?

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 a square root function. Let we define the given function as $f(x) = \sqrt x$ . Now we are going to apply the limit to the function $f(x)$ so we have
$\mathop {\lim }\limits_{x \to \infty } f(x) = \mathop {\lim }\limits_{x \to \infty } \sqrt x$
The function $f(x)$ is a square root function. As x approaches to $\infty$ 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
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } f(x) = \sqrt \infty$
The square root of infinity is an infinity so we have
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } f(x) = \infty$
Hence the when the x approaches to infinity of a square root function
Therefore we have found the limit for the function $f(x) = \sqrt x$
Hence $\mathop {\lim }\limits_{x \to \infty } \sqrt x = \infty$
So, the correct answer is “$\infty$”.

Note : The square root of the function is defined as $f(x) = \sqrt x$ , where x is an unknown variable. When the x approaches to infinity, it implies that function is going to the infinity and the value will be infinity. Infinity is a large number and doesn't know the exact value. Therefore the square root of an unknown greater number will be infinity.