009C Sample Final 3, Problem 1 - Revision history

Kayla Murray at 01:39, 3 December 2017

2017-12-03T01:39:55Z

2017-03-06T00:49:40Z

2017-03-06T00:46:46Z

2017-03-06T00:33:11Z

2017-03-06T00:25:44Z

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2017-03-05T01:17:55Z

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2017-02-18T18:39:28Z

@@ Line 138: / Line 138: @@
 & = & \displaystyle{\lim_{x\rightarrow \infty} \frac{\ln \bigg(\frac{1+x}{x}\bigg)}{\frac{1}{x}}}\\
 &&\\
-& \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow \infty} \frac{\frac{x}{1+x}\frac{-1}{x^2}}{\big(-\frac{1}{x^2}\big)}}\\
+& \overset{L'H}{=} & \displaystyle{\lim_{x\rightarrow \infty} \frac{\frac{x}{1+x}\big(\frac{-1}{x^2}\big)}{\big(-\frac{1}{x^2}\big)}}\\
 &&\\
 & = & \displaystyle{\lim_{x\rightarrow \infty} \frac{x}{1+x}}\\
@@ Line 149: / Line 149: @@
 !Step 4: &nbsp;
 |-
-|Since &nbsp;<math>\ln y= 1,</math>&nbsp; we know
+|Since &nbsp;<math style="vertical-align: -4px">\ln y= 1,</math>&nbsp; we know
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>y=e.</math>

@@ Line 90: / Line 90: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
 |-
+|
 |-
+|Now, we have
 |-
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
+!Step 4: &nbsp;
 |-
+|we have
 |-
+|
 |}
@@ Line 112: / Line 173: @@
 |&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp;<math>e^{\frac{1}{2}}</math>
 |-
-|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp;
+|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp;<math>\text{DNE}</math>
 |}
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 57: / Line 57: @@
 |-
 |Hence, we can use L'Hopital's Rule to calculate this limit.
 |-
+|
 |}
@@ Line 85: / Line 110: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp;&nbsp; '''(a)'''
+|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp;<math>e^{\frac{1}{2}}</math>
 |-
-|&nbsp;&nbsp; '''(b)'''
+|&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp;
 |}
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 28: / Line 28: @@
 !Step 1: &nbsp;
 |-
+|Let
 |-
+|We then take the natural log of both sides to get
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\ln y = \ln\bigg(\lim_{n\rightarrow \infty} \bigg(1+\frac{1}{2n}\bigg)^n\bigg).</math>
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 2: &nbsp;
 |-
+|
 |-
+|

@@ Line 1: / Line 1: @@
-<span class="exam"> Which of the following sequences &nbsp;<math>(a_n)_{n\ge 1}</math>&nbsp; converges? Which diverges? Give reasons for your answers!
+<span class="exam"> Which of the following sequences &nbsp;<math style="vertical-align: -5px">(a_n)_{n\ge 1}</math>&nbsp; converges? Which diverges? Give reasons for your answers!
-<span class="exam">(a) &nbsp;<math>a_n=\bigg(1+\frac{1}{2n}\bigg)^n</math>
+<span class="exam">(a) &nbsp;<math style="vertical-align: -15px">a_n=\bigg(1+\frac{1}{2n}\bigg)^n</math>
-<span class="exam">(b) &nbsp;<math>a_n=\cos(n\pi)\bigg(\frac{1+n}{n}\bigg)^n</math>
+<span class="exam">(b) &nbsp;<math style="vertical-align: -15px">a_n=\cos(n\pi)\bigg(\frac{1+n}{n}\bigg)^n</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 8: / Line 8: @@
 !Foundations: &nbsp;
 |-
+|'''L'Hôpital's Rule'''
 |-
+|
 |-
+|
 |-
+|
 |}

@@ Line 1: / Line 1: @@
-<span class="exam">Compute
+<span class="exam"> Which of the following sequences <math>(a_n)_{n\ge 1}</math> converges? Which diverges? Give reasons for your answers!
-::<span class="exam">a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math>
+::<span class="exam">a)&nbsp; <math>a_n=\bigg(1+\frac{1}{2n}\bigg)^n</math>
-::<span class="exam">b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math>
+::<span class="exam">b)&nbsp; <math>a_n=\cos(n\pi)\bigg(\frac{1+n}{n}\bigg)^n</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"