Finding a Limit

Find:

1. \(\lim\limits_{x\rightarrow 1}[2x(x+4)]\)
2. \(\lim\limits_{x\rightarrow 2^{+}}[4x(2-x)]\)

Solution

1. \[ \begin{array}{rcl@{\qquad}l} \lim\limits_{x\rightarrow 1}[(2x)(x+4)] &=&\left[ \lim\limits_{\kern.5ptx\rightarrow 1}(2x)\right] \left[ \lim\limits_{\kern.5ptx\rightarrow 1}(x+4)\right] &{\color{#0066A7}{\hbox{Limit of a Product}}}\\ &=&\left[ 2\cdot \lim\limits_{x\rightarrow 1}x\right] \cdot \left[ \lim\limits_{\kern.5ptx\rightarrow 1}x+\lim\limits_{x\rightarrow 1}4\right] &{\color{#0066A7}{\hbox{Limit of a Constant Times a Function, Limit of a Sum}}}\\ &=&(2\cdot 1)\cdot (1+4)=10 & {\color{#0066A7}{\hbox{Use (2) and (1), and simplify.}}} \end{array} \]
2. We use properties of limits to find the one-sided limit. \[ \begin{eqnarray*} \lim \limits_{x\rightarrow 2^{+}}[4x(2-x)] &=& 4\lim \limits_{x\rightarrow 2^{+}} [ x( 2-x) ] =4 \Big[ \lim \limits_{x\rightarrow 2^{+}}x \Big ]~ \Big[ \lim \limits_{x\rightarrow 2^{+}}( 2-x) \Big ]\\ &=& 4 \cdot 2 \Big[ \lim \limits_{x \rightarrow 2^{+}}2 - \lim \limits_{x \rightarrow 2^{+}}x \Big ] = 4 \cdot 2 \cdot (2 - 2) = 0 \end{eqnarray*} \]