Question

a. make two tables, one showing the values of g for t = 48.9, 48.99, and 48.999 and one showing values of g for t = 49.1, 49.01, 49.001. b. make a conjecture about the value of \\( \lim_{t \to 49} \frac{t - 49}{\sqrt{t} - 7} \\). a. make a table showing the values of g for t = 48.9, 48.99, and 48.999. (round to four decimal places.) \

$$\begin{tabular}{|c|c|} \\hline t & g(t) \\\\ \\hline 48.9 & 13.9929 \\\\ \\hline 48.99 & 13.9993 \\\\ \\hline 48.999 & 13.9999 \\\\ \\hline \\end{tabular}$$

make a table showing the values of g for t = 49.1, 49.01, and 49.001. (round to four decimal places.) \

$$\begin{tabular}{|c|c|} \\hline t & g(t) \\\\ \\hline 49.1 & \\square \\\\ \\hline 49.01 & \\square \\\\ \\hline 49.001 & \\square \\\\ \\hline \\end{tabular}$$

Explanation:

The function here is \( g(t)=\frac{t - 49}{\sqrt{t}-7} \). We can simplify this function by rationalizing the denominator. Multiply the numerator and denominator by \( \sqrt{t}+7 \):

\[

$$\begin{align*} g(t)&=\frac{(t - 49)(\sqrt{t}+7)}{(\sqrt{t}-7)(\sqrt{t}+7)}\\ &=\frac{(t - 49)(\sqrt{t}+7)}{t - 49} \quad (\text{since } (\sqrt{t})^2-7^2=t - 49)\\ &=\sqrt{t}+7 \quad (\text{for } t eq49) \end{align*}$$

\]

Now we can use this simplified function to find the values of \( g(t) \) for the given \( t \) values.

Step 1: For \( t = 49.1 \)

Substitute \( t = 49.1 \) into \( g(t)=\sqrt{t}+7 \):
\( g(49.1)=\sqrt{49.1}+7 \)
Calculate \( \sqrt{49.1}\approx7.00714 \) (using a calculator), then \( g(49.1)\approx7.00714 + 7=14.00714\approx14.0071 \) (rounded to four decimal places).

Step 2: For \( t = 49.01 \)

Substitute \( t = 49.01 \) into \( g(t)=\sqrt{t}+7 \):
\( g(49.01)=\sqrt{49.01}+7 \)
Calculate \( \sqrt{49.01}\approx7.000714 \), then \( g(49.01)\approx7.000714+7 = 14.000714\approx14.0007 \) (rounded to four decimal places).

Step 3: For \( t = 49.001 \)

Substitute \( t = 49.001 \) into \( g(t)=\sqrt{t}+7 \):
\( g(49.001)=\sqrt{49.001}+7 \)
Calculate \( \sqrt{49.001}\approx7.0000714 \), then \( g(49.001)\approx7.0000714 + 7=14.0000714\approx14.0001 \) (rounded to four decimal places).

Answer:

For \( t = 49.1 \), \( g(t)\approx14.0071 \); for \( t = 49.01 \), \( g(t)\approx14.0007 \); for \( t = 49.001 \), \( g(t)\approx14.0001 \)

The table is:

\( t \)	\( g(t) \)
49.01	14.0007
49.001	14.0001