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Answer added to 1.5 Limits with Infinite Inputs (LT5) except 1.58 (b) #511

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Answer added to 1.5 Limits with Infinite Inputs (LT5) except 1.58 (b) #511

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150 changes: 150 additions & 0 deletions source/calculus/source/01-LT/05.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -60,6 +60,11 @@
</p>
</li>
</ol>
<answer>
<p>
D. As <m>x</m> gets smaller, the function <m>x^3</m> gets smaller and smaller.
</p>
</answer>
</activity>

<remark>
Expand Down Expand Up @@ -125,6 +130,11 @@
</p>
</li>
</ol>
<answer>
<p>
B. As <m>x</m> tends to <m>-\infty</m>, the function <m>1/x^3</m> tends to 0.
</p>
</answer>
</activity>

<definition>
Expand Down Expand Up @@ -255,6 +265,11 @@
</figure>
</sidebyside>
</introduction>
<answer>
<p>
C, D and E
</p>
</answer>
</activity>

<activity xml:id="infinity-rational1">
Expand Down Expand Up @@ -290,6 +305,11 @@
</p>
</li>
</ol>
<answer>
<p>
D. The only possible limits are any constant number or <m>\pm \infty</m>..
</p>
</answer>
</activity>

<activity xml:id="infinity-rational2">
Expand Down Expand Up @@ -335,6 +355,12 @@
</p>
</li>
</ol>
<answer>
<p> A. <m> \frac{1}{2} </m> </p>

<p> D. <m> 2 </m> </p>
<p> E. <m> -4 </m> </p>
</answer>
</task>


Expand Down Expand Up @@ -385,6 +411,11 @@
</p>
</li>
</ol>
<answer>
<p>
C. When the degree of the numerator is less than the degree of the denominator.
</p>
</answer>
</task>


Expand Down Expand Up @@ -419,6 +450,11 @@
</p>
</li>
</ol>
<answer>
<p>
B. When the degree of the numerator is greater than the degree of the denominator.
</p>
</answer>
</task>


Expand All @@ -428,6 +464,15 @@
Test your rules by creating a rational function whose limit as <m>x \to \infty</m> equals 0 and another whose limit as <m>x \to \infty</m> is infinite.
Then check them graphically.
</p>
<answer>

<p>
<m>\displaystyle \lim_{x\to \infty} \dfrac{x^2-x+3}{ 2x^3-3x+5} = 0 </m>
</p>
<p>
<m>\displaystyle \lim_{x\to \infty} \dfrac{2x^3-3x^2+5}{ 5x^2-x+1} </m> is infinite.
</p>
</answer>
</task>
</activity>

Expand All @@ -445,6 +490,17 @@
\lim_{x\to-\infty } -\dfrac{6 \, {x^4} + 7 \, {x^3} - 7}{6 \, x - {x^4} + 9} \,\text{ and }\, \lim_{x\to+\infty } -\dfrac{6 \, {x^4} + 7 \, {x^3} - 7}{6 \, x - {x^4} + 9}
</me>
</p>
<answer>



<p>
<me>
\lim_{x\to-\infty } -\dfrac{6 \, {x^4} + 7 \, {x^3} - 7}{6 \, x - {x^4} + 9} = 6 \,\text{ and }\, \lim_{x\to+\infty } -\dfrac{6 \, {x^4} + 7 \, {x^3} - 7}{6 \, x - {x^4} + 9} = 6
</me>
</p>

</answer>
</task>


Expand All @@ -454,6 +510,13 @@
\lim_{x\to-\infty } -\dfrac{7 \, {x^4} - 5 \, {x^3} + 8}{3 \, {\left(2 \, {x^5} + 3 \, {x^2} - 3\right)}} \,\text{ and }\, \lim_{x\to+\infty } -\dfrac{7 \, {x^4} - 5 \, {x^3} + 8}{3 \, {\left(2 \, {x^5} + 3 \, {x^2} - 3\right)}}
</me>
</p>
<answer>
<p>
<me>
\lim_{x\to-\infty } -\dfrac{7 \, {x^4} - 5 \, {x^3} + 8}{3 \, {\left(2 \, {x^5} + 3 \, {x^2} - 3\right)}} = 0 \,\text{ and }\, \lim_{x\to+\infty } -\dfrac{7 \, {x^4} - 5 \, {x^3} + 8}{3 \, {\left(2 \, {x^5} + 3 \, {x^2} - 3\right)}} = 0
</me>
</p>
</answer>
</task>


Expand All @@ -463,6 +526,13 @@
\lim_{x\to-\infty } \dfrac{3 \, {x^6} + {x^3} - 8}{7 \, x - 6 \, {x^5} + 7} \,\text{ and }\, \lim_{x\to+\infty } \dfrac{3 \, {x^6} + {x^3} - 8}{7 \, x - 6 \, {x^5} + 7}
</me>
</p>
<answer>
<p>
<me>
\lim_{x\to-\infty } \dfrac{3 \, {x^6} + {x^3} - 8}{7 \, x - 6 \, {x^5} + 7} = +\infty \,\text{ and }\, \lim_{x\to+\infty } \dfrac{3 \, {x^6} + {x^3} - 8}{7 \, x - 6 \, {x^5} + 7} = -\infty
</me>
</p>
</answer>
</task>
<!--
<answer>
Expand Down Expand Up @@ -545,6 +615,11 @@
</p>
</li>
</ol>
<answer>
<p>
E. The limit does not exist.
</p>
</answer>
</activity>

<warning>
Expand All @@ -565,41 +640,71 @@
<p>
<m>\displaystyle \lim_{x\to -\infty} \dfrac{x^3-x+83}{1} </m>
</p>
<answer>
<p>
<m>-\infty</m>
</p>
</answer>
</task>


<task>
<p>
<m>\displaystyle \lim_{x\to -\infty} \dfrac{1}{x^3-x+83} </m>
</p>
<answer>
<p>
0
</p>
</answer>
</task>


<task>
<p>
<m>\displaystyle \lim_{x\to +\infty} \dfrac{x+3}{2-x}</m>
</p>
<answer>
<p>
-1
</p>
</answer>
</task>


<task>
<p>
<m>\displaystyle \lim_{x\to -\infty} \dfrac{\pi-3x}{\pi x-3}</m>
</p>
<answer>
<p>
<m>\frac{-3}{\pi}</m>
</p>
</answer>
</task>


<task>
<p>
(Challenge) <m>\displaystyle \lim_{x\to + \infty} \dfrac{3e^x+2}{2e^x+3}</m>
</p>
<answer>
<p>
<m>\frac{3}{2}</m>
</p>
</answer>
</task>


<task>
<p>
(Challenge) <m>\displaystyle \lim_{x\to - \infty} \dfrac{3e^x+2}{2e^x+3}</m>
</p>
<answer>
<p>
<m>\frac{2}{3}</m>
</p>
</answer>
</task>
</activity>

Expand Down Expand Up @@ -633,6 +738,11 @@
Then, prove that your guess is right using algebra.
</p>
</statement>
<answer>
<p>
<m> f(x)= 2</m>
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Should this be y = 2?

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Suggested change
<m> f(x)= 2</m>
<m> y= 2</m>

</p>
</answer>
</task>


Expand All @@ -642,6 +752,11 @@
Use limit notation to describe the behavior of <m>f(x)</m> at its horizontal asymptotes.
</p>
</statement>
<answer>
<p>
<m> \displaystyle \lim_{x \rightarrow \infty }f(x)= 2</m>
</p>
</answer>
</task>
<!--
<task>
Expand All @@ -660,6 +775,11 @@
Come up with the formula of a rational function that has horizontal asymptote <m>y=3</m>.
</p>
</statement>
<answer>
<p>
<m> \displaystyle \lim_{x \rightarrow \infty }\frac{3x^2 -x +1 }{x^2+2 }</m>
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I think either we need to say "here is one possible answer" or leave no answer here, since the prompt is to construct a function.

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Suggested change
<m> \displaystyle \lim_{x \rightarrow \infty }\frac{3x^2 -x +1 }{x^2+2 }</m>
One possible answer: <m> \displaystyle \lim_{x \rightarrow \infty }\frac{3x^2 -x +1 }{x^2+2 }</m>

</p>
</answer>
</task>


Expand All @@ -669,6 +789,11 @@
What do you think is happening around <m>x=3</m>? We will come back to this in the next section!
</p>
</statement>
<answer>
<p>
Around <m>x= 3 </m>, <m>f(x)</m> is either going toward <m>- \infty </m> or <m> \infty </m>
</p>
</answer>
</task>
</activity>
<!-- application activity -->
Expand Down Expand Up @@ -702,6 +827,11 @@
In the long run, what temperature do you expect the coffee to tend to? Write your observation with limit notation.
</p>
</statement>
<answer>
<p>
<m> \lim_{ t \rightarrow \infty}Q(t) = 72 </m> degrees Fahrenheit, where <m>Q(t)</m> is the temperature of coffee at anytime <m>t</m>.
</p>
</answer>
</task>


Expand All @@ -712,6 +842,11 @@
Which one?
</p>
</statement>
<answer>
<p>
<m>c = 72 </m>
</p>
</answer>
</task>


Expand All @@ -723,6 +858,11 @@
Use this to find the value of <m>b</m> for the exponential model described in this scenario.
</p>
</statement>
<answer>
<p>
<m>b = 0.9 </m>
</p>
</answer>
</task>


Expand All @@ -733,6 +873,11 @@
Use the data about the initial temperature to find the value of the parameter <m>a</m> in the model <m> Q(t) = a \, b^t + c</m>.
</p>
</statement>
<answer>
<p>
<m> Q(t) = 100 (0.9)^t + 72 </m>.
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This should be 28(0.9)^t + 72

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Suggested change
<m> Q(t) = 100 (0.9)^t + 72 </m>.
<m> Q(t) = 28(0.9)^t + 72 </m>.

</p>
</answer>
</task>


Expand All @@ -743,6 +888,11 @@
If you go back to drink the cup of coffee 30 minutes after it was left on the counter, what temperature will the coffee have reached?
</p>
</statement>
<answer>
<p>
<m> Q(30) = 76.23 </m>.
</p>
</answer>
</task>
</activity>
</subsection>
Expand Down
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