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### Section 2.4 : Limit Properties

3. Given $$\mathop {\lim }\limits_{x \to 0} f\left( x \right) = 6$$, $$\mathop {\lim }\limits_{x \to 0} g\left( x \right) = - 4$$ and $$\mathop {\lim }\limits_{x \to 0} h\left( x \right) = - 1$$ use the limit properties given in this section to compute each of the following limits. If it is not possible to compute any of the limits clearly explain why not.

1. $$\mathop {\lim }\limits_{x \to 0} {\left[ {f\left( x \right) + h\left( x \right)} \right]^3}$$
2. $$\mathop {\lim }\limits_{x \to 0} \sqrt {g\left( x \right)h\left( x \right)}$$
3. $$\mathop {\lim }\limits_{x \to 0} \sqrt{{11 + {{\left[ {g\left( x \right)} \right]}^2}}}$$
4. $$\displaystyle \mathop {\lim }\limits_{x \to 0} \sqrt {\frac{{f\left( x \right)}}{{h\left( x \right) - g\left( x \right)}}}$$
Hint : For each of these all we need to do is use the limit properties on the limit until the given limits appear and we can then compute the value.

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a $$\mathop {\lim }\limits_{x \to 0} {\left[ {f\left( x \right) + h\left( x \right)} \right]^3}$$ Show Solution

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*}

b $$\mathop {\lim }\limits_{x \to 0} \sqrt {g\left( x \right)h\left( x \right)}$$ Show Solution

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*}

c $$\mathop {\lim }\limits_{x \to 0} \sqrt{{11 + {{\left[ {g\left( x \right)} \right]}^2}}}$$ Show Solution

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{{11 + {{\left[ {g\left( x \right)} \right]}^2}}} & = \sqrt{{\mathop {\lim }\limits_{x \to 0} \left( {11 + {{\left[ {g\left( x \right)} \right]}^2}} \right)}} & & \hspace{0.25in}{\mbox{Property 6}}\\ & = \sqrt{{\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{{\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{{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*}

d $$\displaystyle \mathop {\lim }\limits_{x \to 0} \sqrt {\frac{{f\left( x \right)}}{{h\left( x \right) - g\left( x \right)}}}$$ Show Solution

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.