Answer to Question #304043 in Calculus for Luna Heart

Question

Construct a table of values to investigate the following limits.

1. lim (10/x-2)

x-> 7

2. lim (x+6/x+2)

x-> 1

3. lim f(x)

x-> -1

If f(x) { 1/x if x ≤ -1}

If f(x) { x2-2 if x > -1}

Answer 1

1. The following table gives values of for values of close to $7,$ but not

equal to $7.$

$\def\arraystretch{1.5} \begin{array}{c:c} x & f(x) \\ \hline 6 & 2.500000 \\ \hdashline 6.5 & 2.222222 \\ \hdashline 6.9 & 2.040816\\ \hdashline 6.99 & 2.004008 \\ \hdashline 6.999 & 2.000400 \\ \hdashline 6.9999 & 2.000040 \\ \hdashline 7.0001 & 1.999960 \\ \hdashline 7.001 & 1.999600 \\ \hdashline 7.01 & 1.996008 \\ \hdashline 7.1 & 1.960784 \\ \hdashline 7.5 & 1.818182 \\ \hdashline 8 & 1.666667 \\ \hdashline \end{array}$

$\lim\limits_{x\to 7}\dfrac{10}{x-2}=2$

2. The following table gives values of for values of close to $2,$ but not

equal to $1.$

$\def\arraystretch{1.5} \begin{array}{c:c} x & f(x) \\ \hline 0 & 3.000000 \\ \hdashline 0.5 & 2.600000 \\ \hdashline 0.9 & 2.379310\\ \hdashline 0.99 & 2.337793 \\ \hdashline 0.999 & 2.333778 \\ \hdashline 0.9999 & 2.333378 \\ \hdashline 1.0001 & 2.333289 \\ \hdashline 1.001 & 2.332889 \\ \hdashline 1.01 & 2.328904 \\ \hdashline 1.1 & 2.290323 \\ \hdashline 1.5 & 2.142857 \\ \hdashline 2 & 2.000000 \\ \hdashline \end{array}$

$\lim\limits_{x\to 1}\dfrac{x+6}{x+2}=\dfrac{7}{3}$

3. The following table gives values of for values of close to $-1,$ but not

equal to $-1.$

$\def\arraystretch{1.5} \begin{array}{c:c} x & f(x) \\ \hline -2 & -0.500000 \\ \hdashline -1.5 & -0.666667 \\ \hdashline -1.1 & -0.909091\\ \hdashline -1.01.99 & -0.990099 \\ \hdashline -1.001 & -0.999001\\ \hdashline -1.0001 & -0.999900 \\ \hdashline -0.9999 & -1.000200 \\ \hdashline -0.999 & -1.001999 \\ \hdashline -0.99 & -1.0199\\ \hdashline -0.9 & -1.19 \\ \hdashline -0.5 & -1.75 \\ \hdashline 0 & -2 \\ \hdashline \end{array}$

$\lim\limits_{x\to -1}f(x)=-1$