Question

let ( g(t) = \frac{t - 49}{sqrt{t} - 7} ).
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 ), and ( 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)
\\( \

$$\begin{array}{c|c} t & g(t) \\\\ \\hline 48.9 & \\square \\\\ 48.99 & \\square \\\\ 48.999 & \\square \\\\ \\end{array}$$

\\)

Explanation:

Part a:

Step 1: For \( t = 48.9 \)

We have the function \( g(t)=\frac{t - 49}{\sqrt{t}-7} \). First, simplify the function by rationalizing the denominator. Multiply the numerator and denominator by \( \sqrt{t}+7 \):
\[

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

\]
Now, substitute \( t = 48.9 \) into \( \sqrt{t}+7 \):
\( \sqrt{48.9}+7\approx6.992857 + 7=13.992857\approx13.9929 \) (rounded to four decimal places)

Step 2: For \( t = 48.99 \)

Substitute \( t = 48.99 \) into \( \sqrt{t}+7 \):
\( \sqrt{48.99}+7\approx6.999286+7 = 13.999286\approx13.9993 \) (rounded to four decimal places)

Step 3: For \( t = 48.999 \)

Substitute \( t = 48.999 \) into \( \sqrt{t}+7 \):
\( \sqrt{48.999}+7\approx6.999929+7=13.999929\approx13.9999 \) (rounded to four decimal places)

Part b:

From the table in part a, as \( t \) approaches \( 49 \) from the left (\( t = 48.9,48.99,48.999 \)), the values of \( g(t) \) are \( 13.9929,13.9993,13.9999 \) respectively, which are approaching \( 14 \).

If we consider the right - hand limit (for \( t = 49.1,49.01,49.001 \)):
For \( t = 49.1 \), \( \sqrt{49.1}+7\approx7.007134 + 7=14.007134\approx14.0071 \)
For \( t = 49.01 \), \( \sqrt{49.01}+7\approx7.000714+7 = 14.000714\approx14.0007 \)
For \( t = 49.001 \), \( \sqrt{49.001}+7\approx7.000071+7 = 14.000071\approx14.0001 \)

As \( t \) approaches \( 49 \) from both sides, the values of \( g(t) \) approach \( 14 \). So we conjecture that \( \lim_{t
ightarrow49}\frac{t - 49}{\sqrt{t}-7}=14 \)

Part a Table:

\( t \)	\( g(t) \)
\( 48.99 \)	\( 13.9993 \)
\( 48.999 \)	\( 13.9999 \)

Part b Answer:

By analyzing the values of \( g(t) \) as \( t \) approaches \( 49 \) from both the left (using \( t = 48.9,48.99,48.999 \)) and the right (using \( t = 49.1,49.01,49.001 \)), we observe that the values of \( g(t) \) approach \( 14 \).

Answer:

\( \lim_{t
ightarrow49}\frac{t - 49}{\sqrt{t}-7}=\boxed{14} \)