Question

let ( g(t) = \frac{t - 100}{sqrt{t} - 10} )

a. make two tables, one showing the values of ( g ) for ( t = 99.9, 99.99, 99.999 ) and one showing values of ( g ) for ( t = 100.1, 100.01, 100.001 ).

b. make a conjecture about the value of ( lim_{t \to 100} \frac{t - 100}{sqrt{t} - 10} ).

a. make a table showing the values of ( g ) for ( t = 99.9, 99.99, 99.999 ) (round to four decimal places).

( t )	( g(t) )
99.99
99.999

make a table showing the values of ( g ) for ( t = 100.1, 100.01, 100.001 ) (round to four decimal places).

( t )	( g(t) )
100.01
100.001

b. make a conjecture about the value of ( lim_{t \to 100} \frac{t - 100}{sqrt{t} - 10} ). select the correct choice below and fill in any answer boxes in your choice.

a. ( lim_{t \to 100} \frac{t - 100}{sqrt{t} - 10} = ) (simplify your answer)

Explanation:

Part a (Left - hand limit, \( t \) approaching 100 from the left: \( t = 99.9, 99.99, 99.999 \))

Step 1: For \( t = 99.9 \)

We use the function \( g(t)=\frac{t - 100}{\sqrt{t}-10} \). First, calculate \( \sqrt{99.9}\approx9.99499875 \)
Then \( g(99.9)=\frac{99.9 - 100}{9.99499875-10}=\frac{- 0.1}{-0.00500125}\approx19.9950 \) (rounded to four decimal places)

Step 2: For \( t = 99.99 \)

Calculate \( \sqrt{99.99}\approx9.9994999875 \)
Then \( g(99.99)=\frac{99.99 - 100}{9.9994999875-10}=\frac{-0.01}{-0.0005000125}\approx19.9995 \) (rounded to four decimal places)

Step 3: For \( t = 99.999 \)

Calculate \( \sqrt{99.999}\approx9.999949999875 \)
Then \( g(99.999)=\frac{99.999 - 100}{9.999949999875-10}=\frac{-0.001}{-0.000050000125}\approx19.99995 \approx20.0000\) (rounded to four decimal places)

Part a (Right - hand limit, \( t \) approaching 100 from the right: \( t = 100.1, 100.01, 100.001 \))

Step 1: For \( t = 100.1 \)

Calculate \( \sqrt{100.1}\approx10.00499875 \)
Then \( g(100.1)=\frac{100.1 - 100}{10.00499875-10}=\frac{0.1}{0.00499875}\approx20.0050 \) (rounded to four decimal places)

Step 2: For \( t = 100.01 \)

Calculate \( \sqrt{100.01}\approx10.0004999875 \)
Then \( g(100.01)=\frac{100.01 - 100}{10.0004999875-10}=\frac{0.01}{0.0004999875}\approx20.0005 \) (rounded to four decimal places)

Step 3: For \( t = 100.001 \)

Calculate \( \sqrt{100.001}\approx10.000049999875 \)
Then \( g(100.001)=\frac{100.001 - 100}{10.000049999875-10}=\frac{0.001}{0.000049999875}\approx20.00005\approx20.0000 \) (rounded to four decimal places)

Part b: Conjecture about the limit

As \( t \) approaches 100 from both the left (values like 99.9, 99.99, 99.999) and the right (values like 100.1, 100.01, 100.001), the values of \( g(t) \) are approaching 20. So we conjecture that \( \lim_{t
ightarrow100}\frac{t - 100}{\sqrt{t}-10}=20 \)

Part c: Analytical solution (Simplify the function)

We can rationalize the denominator of \( g(t)=\frac{t - 100}{\sqrt{t}-10} \). Multiply the numerator and denominator by \( \sqrt{t}+10 \):

\[

$$\begin{align*} g(t)&=\frac{(t - 100)(\sqrt{t}+10)}{(\sqrt{t}-10)(\sqrt{t}+10)}\\ &=\frac{(t - 100)(\sqrt{t}+10)}{t - 100}\\ &=\sqrt{t}+10\quad (t eq100) \end{align*}$$

\]

Now, find the limit as \( t
ightarrow100 \):

\[
\lim_{t
ightarrow100}g(t)=\lim_{t
ightarrow100}(\sqrt{t}+10)=\sqrt{100}+10 = 10 + 10=20
\]

Tables:

Left - hand limit table (\( t

ightarrow100^- \))

\( t \)	\( g(t) \) (rounded to 4 decimals)
99.99	19.9995
99.999	20.0000

Right - hand limit table (\( t

ightarrow100^+ \))

\( t \)	\( g(t) \) (rounded to 4 decimals)
100.01	20.0005
100.001	20.0000

Final Answer for the limit:

\(\boxed{20}\)

Answer:

Part a (Left - hand limit, \( t \) approaching 100 from the left: \( t = 99.9, 99.99, 99.999 \))

Step 1: For \( t = 99.9 \)

We use the function \( g(t)=\frac{t - 100}{\sqrt{t}-10} \). First, calculate \( \sqrt{99.9}\approx9.99499875 \)
Then \( g(99.9)=\frac{99.9 - 100}{9.99499875-10}=\frac{- 0.1}{-0.00500125}\approx19.9950 \) (rounded to four decimal places)

Step 2: For \( t = 99.99 \)

Calculate \( \sqrt{99.99}\approx9.9994999875 \)
Then \( g(99.99)=\frac{99.99 - 100}{9.9994999875-10}=\frac{-0.01}{-0.0005000125}\approx19.9995 \) (rounded to four decimal places)

Step 3: For \( t = 99.999 \)

Calculate \( \sqrt{99.999}\approx9.999949999875 \)
Then \( g(99.999)=\frac{99.999 - 100}{9.999949999875-10}=\frac{-0.001}{-0.000050000125}\approx19.99995 \approx20.0000\) (rounded to four decimal places)

Part a (Right - hand limit, \( t \) approaching 100 from the right: \( t = 100.1, 100.01, 100.001 \))

Step 1: For \( t = 100.1 \)

Calculate \( \sqrt{100.1}\approx10.00499875 \)
Then \( g(100.1)=\frac{100.1 - 100}{10.00499875-10}=\frac{0.1}{0.00499875}\approx20.0050 \) (rounded to four decimal places)

Step 2: For \( t = 100.01 \)

Calculate \( \sqrt{100.01}\approx10.0004999875 \)
Then \( g(100.01)=\frac{100.01 - 100}{10.0004999875-10}=\frac{0.01}{0.0004999875}\approx20.0005 \) (rounded to four decimal places)

Step 3: For \( t = 100.001 \)

Calculate \( \sqrt{100.001}\approx10.000049999875 \)
Then \( g(100.001)=\frac{100.001 - 100}{10.000049999875-10}=\frac{0.001}{0.000049999875}\approx20.00005\approx20.0000 \) (rounded to four decimal places)

Part b: Conjecture about the limit

As \( t \) approaches 100 from both the left (values like 99.9, 99.99, 99.999) and the right (values like 100.1, 100.01, 100.001), the values of \( g(t) \) are approaching 20. So we conjecture that \( \lim_{t
ightarrow100}\frac{t - 100}{\sqrt{t}-10}=20 \)

Part c: Analytical solution (Simplify the function)

We can rationalize the denominator of \( g(t)=\frac{t - 100}{\sqrt{t}-10} \). Multiply the numerator and denominator by \( \sqrt{t}+10 \):

\[

$$\begin{align*} g(t)&=\frac{(t - 100)(\sqrt{t}+10)}{(\sqrt{t}-10)(\sqrt{t}+10)}\\ &=\frac{(t - 100)(\sqrt{t}+10)}{t - 100}\\ &=\sqrt{t}+10\quad (t eq100) \end{align*}$$

\]

Now, find the limit as \( t
ightarrow100 \):

\[
\lim_{t
ightarrow100}g(t)=\lim_{t
ightarrow100}(\sqrt{t}+10)=\sqrt{100}+10 = 10 + 10=20
\]

Tables:

Left - hand limit table (\( t

ightarrow100^- \))

\( t \)	\( g(t) \) (rounded to 4 decimals)
99.99	19.9995
99.999	20.0000

Right - hand limit table (\( t

ightarrow100^+ \))

\( t \)	\( g(t) \) (rounded to 4 decimals)
100.01	20.0005
100.001	20.0000

Final Answer for the limit:

\(\boxed{20}\)