Chapter 11 - Section 11.1 - Finding Limits Using Tables and Graphs - Exercise Set - Page 1138: 9

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Consider the provided limit notation, $\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}$. In making the table, choose the value of x close to 0 from the left and from the right as x approaches 0. As x approaches 0 from the left, arbitrarily start with $ x=-0.01$. Then select two additional values of x that are closer to 0, but still less than 0; we choose $-0.001$ and $-0.0001$. Evaluate f at each chosen value of x to obtain the corresponding values of $ f\left( x \right)$. At $ x=-0.01$, Substitute, $ x=-0.01$ in the provided limit notation, $\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}$. Therefore, $\begin{align} & \underset{x\to -0.01}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}=\frac{-0.01}{{{\left( -0.01 \right)}^{2}}+1} \\ & =\frac{-0.01}{0.0001+1} \\ & =\frac{-0.01}{1.0001} \\ & =-0.0100 \end{align}$ Now, at $ x=-0.001$ Substitute, $ x=-0.001$ in the provided limit notation, $\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}$. $\begin{align} & \underset{x\to -0.001}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}=\frac{-0.001}{{{\left( -0.001 \right)}^{2}}+1} \\ & =\frac{-0.001}{0.000001+1} \\ & =\frac{-0.01}{1.000001} \\ & =-0.0010 \end{align}$ At $ x=-0.0001$ Substitute, $ x=-0.0001$ in the provided limit notation, $\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}$. $\begin{align} & \underset{x\to -0.0001}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}=\frac{-0.0001}{{{\left( -0.0001 \right)}^{2}}+1} \\ & =\frac{-0.0001}{0.00000001+1} \\ & =\frac{-0.01}{1.00000001} \\ & =-0.0001 \end{align}$ Now, as x approaches 0 from the right, arbitrarily start with $ x=0.01$. Then select two additional values of x that are closer to 0, but still greater than 0; we choose 0.001 and 0.0001. At $ x=0.01$ Substitute, $ x=0.01$ in the provided limit notation, $\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}$. Therefore, $\begin{align} & \underset{x\to 0.01}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}=\frac{0.01}{{{\left( 0.01 \right)}^{2}}+1} \\ & =\frac{0.01}{0.0001+1} \\ & =\frac{0.01}{1.0001} \\ & =0.0100 \end{align}$ Now, at $ x=0.001$ Substitute, $ x=0.001$ in the provided limit notation, $\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}$. Therefore, $\begin{align} & \underset{x\to 0.001}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}=\frac{0.001}{{{\left( 0.001 \right)}^{2}}+1} \\ & =\frac{0.001}{0.000001+1} \\ & =\frac{0.001}{1.000001} \\ & =0.0010 \end{align}$ At $ x=0.0001$ Substitute, $ x=0.0001$ in the provided limit notation, $\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}$. $\begin{align} & \underset{x\to 0.0001}{\mathop{\lim }}\,\frac{x}{{{x}^{2}}+1}=\frac{0.0001}{{{\left( 0.0001 \right)}^{2}}+1} \\ & =\frac{0.0001}{0.00000001+1} \\ & =\frac{0.01}{1.00000001} \\ & =0.0001 \end{align}$ The limit of $\frac{x}{{{x}^{2}}+1}$ as x approaches 0 equals the number 0.