@@ -255,6 +265,11 @@
+
+
+ C, D and E
+
+
@@ -290,6 +305,11 @@
+
+
+ D. The only possible limits are any constant number or \pm \infty..
+
+
@@ -335,6 +355,12 @@
+
+ A. \frac{1}{2}
+
+ D. 2
+ E. -4
+
@@ -385,6 +411,11 @@
+
+
+ C. When the degree of the numerator is less than the degree of the denominator.
+
+
@@ -419,6 +450,11 @@
+
+
+ B. When the degree of the numerator is greater than the degree of the denominator.
+
+
@@ -428,6 +464,15 @@
Test your rules by creating a rational function whose limit as x \to \infty equals 0 and another whose limit as x \to \infty is infinite.
Then check them graphically.
+
+
+
+ \displaystyle \lim_{x\to \infty} \dfrac{x^2-x+3}{ 2x^3-3x+5} = 0
+
+
+ \displaystyle \lim_{x\to \infty} \dfrac{2x^3-3x^2+5}{ 5x^2-x+1} is infinite.
+
+
@@ -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}
+
+
+
+
+
+
+ \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
+
+
+
+
@@ -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)}}
+
+
+
+ \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
+
+
+
@@ -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}
+
+
+
+ \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
+
+
+
@@ -702,6 +827,11 @@
In the long run, what temperature do you expect the coffee to tend to? Write your observation with limit notation.
+
+
+ \lim_{ t \rightarrow \infty}Q(t) = 72 degrees Fahrenheit, where Q(t) is the temperature of coffee at anytime t.
+
+
@@ -712,6 +842,11 @@
Which one?
+
+
+ c = 72
+
+
@@ -723,6 +858,11 @@
Use this to find the value of b for the exponential model described in this scenario.
+
+
+ b = 0.9
+
+
@@ -733,6 +873,11 @@
Use the data about the initial temperature to find the value of the parameter a in the model Q(t) = a \, b^t + c.
+
+
+ Q(t) = 100 (0.9)^t + 72 .
+
+
@@ -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?
+
+
+ Q(30) = 76.23 .
+
+