Answer
The value of $\underset{x\to 3}{\mathop{\lim }}\,\sqrt{{{x}^{2}}-3x+4}$ is $2$.
Work Step by Step
Consider the function $ f\left( x \right)=\sqrt{{{x}^{2}}-3x+4}$,
The function $ g\left( x \right)={{x}^{2}}-3x+4$ is a polynomial.
Find the value of $\underset{x\to 3}{\mathop{\lim }}\,\sqrt{{{x}^{2}}-3x+4}$,
$\begin{align}
& \underset{x\to 3}{\mathop{\lim }}\,\sqrt{{{x}^{2}}-3x+4}=\sqrt{\underset{x\to 3}{\mathop{\lim }}\,\left( {{x}^{2}}-3x+4 \right)} \\
& =\sqrt{\underset{x\to 3}{\mathop{\lim }}\,{{x}^{2}}-3\underset{x\to 3}{\mathop{\lim }}\,x+4} \\
& =\sqrt{{{\left( 3 \right)}^{2}}-3\left( 3 \right)+4} \\
& =\sqrt{9-9+4}
\end{align}$
Further simplify
$\begin{align}
& \underset{x\to 3}{\mathop{\lim }}\,\sqrt{{{x}^{2}}-3x+4}=\sqrt{9-9+4} \\
& =\sqrt{4} \\
& =2
\end{align}$
Thus, the value of $\underset{x\to 3}{\mathop{\lim }}\,\sqrt{{{x}^{2}}-3x+4}$ is $2$.
