Chapter 11 - Section 11.1 - Finding Limits Using Tables and Graphs - Exercise Set - Page 1138: 8

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Consider the provided limit notation, $\underset{x\to 4}{\mathop{\lim }}\,\frac{1}{x-3}$. In making the table, choose the value of x close to 4 from the left and from the right as x approaches 4. As x approaches 4 from the left, arbitrarily start with $ x=3.99$. Then select two additional values of x that are closer to 4, but still less than 4; we choose 3.999 and 3.9999. Evaluate f at each chosen value of x to obtain the corresponding values of $ f\left( x \right)$. At $ x=3.99$, Substitute, $ x=3.99$ in the provided limit notation, $\underset{x\to 4}{\mathop{\lim }}\,\frac{1}{x-3}$. Therefore, $\begin{align} & \underset{x\to 3.99}{\mathop{\lim }}\,\frac{1}{x-3}=\frac{1}{3.99-3} \\ & =\frac{1}{0.99} \\ & =1.0101 \end{align}$ Now, at $ x=3.999$ Substitute, $ x=3.999$ in the provided limit notation, $\underset{x\to 4}{\mathop{\lim }}\,\frac{1}{x-3}$. $\begin{align} & \underset{x\to 3.999}{\mathop{\lim }}\,\frac{1}{x-3}=\frac{1}{3.999-3} \\ & =\frac{1}{0.999} \\ & =1.0010 \end{align}$ At $ x=3.9999$ Substitute, $ x=3.9999$ in the provided limit notation, $\underset{x\to 4}{\mathop{\lim }}\,\frac{1}{x-3}$. $\begin{align} & \underset{x\to 3.9999}{\mathop{\lim }}\,\frac{1}{x-3}=\frac{1}{3.9999-3} \\ & =\frac{1}{0.9999} \\ & =1.0001 \end{align}$ Now, as x approaches 4 from the right, arbitrarily start with $ x=4.01$. Then select two additional values of x that are closer to 4, but still greater than 4; we choose 4.001 and 4.0001. At $ x=4.01$ Substitute, $ x=4.01$ in the provided limit notation, $\underset{x\to 4}{\mathop{\lim }}\,\frac{1}{x-3}$. Therefore, $\begin{align} & \underset{x\to 4.01}{\mathop{\lim }}\,\frac{1}{x-3}=\frac{1}{4.01-3} \\ & =\frac{1}{1.01} \\ & =0.9901 \end{align}$ Now, at $ x=4.001$ Substitute, $ x=4.001$ in the provided limit notation, $\underset{x\to 4}{\mathop{\lim }}\,\frac{1}{x-3}$ Therefore, $\begin{align} & \underset{x\to 4.001}{\mathop{\lim }}\,\frac{1}{x-}=\frac{1}{4.001-3} \\ & =\frac{1}{1.001} \\ & =0.9990 \end{align}$ At $ x=4.0001$ Substitute, $ x=4.0001$ in the provided limit notation, $\underset{x\to 4}{\mathop{\lim }}\,\frac{1}{x-3}$. $\begin{align} & \underset{x\to 4.0001}{\mathop{\lim }}\,\frac{1}{x-3}=\frac{1}{4.0001-3} \\ & =\frac{1}{1.0001} \\ & =0.9999 \end{align}$ The limit of $\frac{1}{x-3}$ as x approaches 4 equals the number 1.