Here is the work for this limit. At each step the property (or properties) used are listed and note that in some cases the properties may have been used more than once in the indicated step.

\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 0} {\left[ {f\left( x \right) + h\left( x \right)} \right]^3} & = {\left[ {\mathop {\lim }\limits_{x \to 0} \left( {f\left( x \right) + h\left( x \right)} \right)} \right]^3} & & \hspace{0.25in}{\mbox{Property 5}}\\ & = {\left[ {\mathop {\lim }\limits_{x \to 0} f\left( x \right) + \mathop {\lim }\limits_{x \to 0} h\left( x \right)} \right]^3} & & \hspace{0.25in}{\mbox{Property 2}}\\ & = {\left[ {6 - 1} \right]^3} & & \hspace{0.25in}{\mbox{Plug in values of limits}}\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{125}} & & & \end{alignat*}\]

Here is the work for this limit. At each step the property (or properties) used are listed and note that in some cases the properties may have been used more than once in the indicated step.

\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 0} \sqrt {g\left( x \right)h\left( x \right)} & = \sqrt {\mathop {\lim }\limits_{x \to 0} g\left( x \right)h\left( x \right)} & & \hspace{0.25in}{\mbox{Property 6}}\\ & = \sqrt {\left[ {\mathop {\lim }\limits_{x \to 0} g\left( x \right)} \right]\left[ {\mathop {\lim }\limits_{x \to 0} h\left( x \right)} \right]} & & \hspace{0.25in}{\mbox{Property 3}}\\ & = \sqrt {\left( { - 4} \right)\left( { - 1} \right)} & & \hspace{0.25in}{\mbox{Plug in value of limits}}\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{2} & & & \end{alignat*}\]

Here is the work for this limit. At each step the property (or properties) used are listed and note that in some cases the properties may have been used more than once in the indicated step.

\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 0} \sqrt[3]{{11 + {{\left[ {g\left( x \right)} \right]}^2}}} & = \sqrt[3]{{\mathop {\lim }\limits_{x \to 0} \left( {11 + {{\left[ {g\left( x \right)} \right]}^2}} \right)}} & & \hspace{0.25in}{\mbox{Property 6}}\\ & = \sqrt[3]{{\mathop {\lim }\limits_{x \to 0} 11 + \mathop {\lim }\limits_{x \to 0} {{\left[ {g\left( x \right)} \right]}^2}}} & & \hspace{0.25in}{\mbox{Property 2}}\\ & = \sqrt[3]{{\mathop {\lim }\limits_{x \to 0} 11 + {{\left[ {\mathop {\lim }\limits_{x \to 0} g\left( x \right)} \right]}^2}}} & & \hspace{0.25in}{\mbox{Property 5}}\\ & = \sqrt[3]{{11 + {{\left( { - 4} \right)}^2}}} & & \hspace{0.25in}{\mbox{Plug in values of limits & Property 7}}\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{3} & & & \end{alignat*}\]

Here is the work for this limit. At each step the property (or properties) used are listed and note that in some cases the properties may have been used more than once in the indicated step.

\[\begin{alignat*}{3}\mathop {\lim }\limits_{x \to 0} \sqrt {\frac{{f\left( x \right)}}{{h\left( x \right) - g\left( x \right)}}} & = \sqrt {\mathop {\lim }\limits_{x \to 0} \frac{{f\left( x \right)}}{{h\left( x \right) - g\left( x \right)}}} & & \hspace{0.25in}{\mbox{Property 6}}\\ & = \sqrt {\frac{{\mathop {\lim }\limits_{x \to 0} f\left( x \right)}}{{\mathop {\lim }\limits_{x \to 0} \left( {h\left( x \right) - g\left( x \right)} \right)}}} & & \hspace{0.25in}{\mbox{Property 4}}\\ & = \sqrt {\frac{{\mathop {\lim }\limits_{x \to 0} f\left( x \right)}}{{\mathop {\lim }\limits_{x \to 0} h\left( x \right) - \mathop {\lim }\limits_{x \to 0} g\left( x \right)}}} & & \hspace{0.25in}{\mbox{Property 2}}\\ & = \sqrt {\frac{6}{{ - 1 - \left( { - 4} \right)}}} & & \hspace{0.25in}{\mbox{Plug in values of limits }}\\ & = \sqrt 2 & & & \end{alignat*}\]

Note that were able to use Property 4 in the second step only because after we evaluated the limit of the denominators (both of them) we found that the limits of the denominators were not zero.