Question

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

Answer 1

Hint: 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.