Question

What is the limit as $x$ approaches the infinity of $\sqrt x$?

Answer 1

Consider the function $f(x) = {x^2}$. Observe that as $x$ takes values very close to $0$, then the value of $f\left( x \right)$ also moves towards $0$ (i.e.)
$\mathop {\lim }\limits_{x \to 0} f(x) = 0$
The limit of $f\left( x \right)$ as $x$ tends to zero is to be thought of as the value $f\left( x \right)$should assume at $x = 0$.
In general as $x \to a,f(x) \to l$, then $l$ is called the limit of the function $f\left( x \right)$ which is written as,
$\mathop {\lim }\limits_{x \to a} f(x) = l$

Complete step-by-step solution:
Consider a function$f(x) = \sqrt x$, $x > 0$
So, we have, $\mathop {\lim }\limits_{x \to \infty } f(x)$ which is $\mathop {\lim }\limits_{x \to \infty } \sqrt x$
Here, we observe that the domain of the function is given to be all positive real numbers i.e. greater than zero. Below we have tabulated the values of the function for positive $x$ (in this table $n$ denotes any positive integer).

$x$	$1$	$100$	$10000$	${10^{2n}}$
$f(x)$	$1$	$10$	$100$	${10^n}$

In the above table, we can see that as $x$ tends to infinity, $f\left( x \right)$ becomes larger and larger and it implies that the value of $f\left( x \right)$ may be made greater than any given number.
Mathematically we can say,
$\mathop {\lim }\limits_{x \to \infty } f(x) = \mathop {\lim }\limits_{x \to \infty } \sqrt x = \infty$
Hence, the limit as $x$ approaches the infinity of $\sqrt x$ is infinity.

Note:

The expected value of the function is determined by the points to the left of a point defining the left hand limit of the function at that point. Similarly the right hand limit can be defined.
Limit of a function at a point is the common value of the left and right hand limits, if they coincide.
In general as $x \to a,f(x) \to l$, then $l$ is called the limit of the function $f\left( x \right)$ which is written as,
$\mathop {\lim }\limits_{x \to a} f(x) = l$
For a function $f$ and a real number $a$, the limit $\mathop {\lim }\limits_{x \to \infty } f(x)$ and $f(a)$ may not be the same.