Calculus and Its Applications, Chapter 1, 1.2, Section 1.2, Problem 2

Determine if the statement
$
\displaystyle
\lim_{x \to 2}
f(x) = 9
$,
then
$
\displaystyle
\lim_{x \to 2}
\sqrt{f(x)} = 3
$.
is either true or false.


The statement is true because according to the limit principles,
$
\displaystyle
\lim_{x \to a} [f(x)]^n
=
\left[ \lim_{x \to a} f(x) \right]^n
=
L^n$
and
$\displaystyle
\lim_{x \to a} \sqrt[n]{f(x)}
=
\sqrt[n]{\lim_{x \to a} f(x)}
=
\sqrt[n]{L}
$


Considering that $n$ is a postive integer and $L \geq 0$. In this case, $n = 2$
$\displaystyle \lim_{x \to 2} f(x) = 9$ and $\displaystyle \lim_{x \to 2} \sqrt{f(x)} = \sqrt{ \lim_{x \to 2} f(x)} = \sqrt{9} = 3$