009A Sample Midterm 2, Problem 1 - Revision history

Kayla Murray at 22:34, 9 November 2017

2017-11-09T22:34:35Z

Kayla Murray at 01:40, 7 November 2017

2017-11-07T01:40:47Z

Kayla Murray at 17:01, 3 November 2017

2017-11-03T17:01:46Z

Kayla Murray at 17:45, 27 March 2017

2017-03-27T17:45:59Z

Kayla Murray at 17:12, 13 March 2017

2017-03-13T17:12:07Z

Kayla Murray at 03:38, 7 March 2017

2017-03-07T03:38:42Z

Kayla Murray at 23:59, 26 February 2017

2017-02-26T23:59:22Z

Kayla Murray at 23:54, 26 February 2017

2017-02-26T23:54:44Z

Kayla Murray at 23:06, 18 February 2017

2017-02-18T23:06:31Z

Kayla Murray at 23:00, 18 February 2017

2017-02-18T23:00:09Z

@@ Line 7: / Line 7: @@
 <span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math>
 <hr>
-[[009A Sample Midterm 2, Problem 1 Detailed Solution|'''<u>Detailed Solution with Background Information</u>''']]
+[[009A Sample Midterm 2, Problem 1 Solution|'''<u>Solution</u>''']]
 [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 7: / Line 7: @@
 <span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math>
 <hr>
-(insert picture of handwritten solution)
+[[File:9ASM2P1.jpg|600px|thumb|center]]
 [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 6: / Line 6: @@
 <span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+[[009A Sample Midterm 2, Problem 1 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
 [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 11: / Line 11: @@
 !Foundations: &nbsp;
 |-
-|<math>\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math>
+|Recall
 |}

@@ Line 102: / Line 102: @@
 |If we look at the graph from the left of &nbsp;<math style="vertical-align: -13px">x=\frac{\pi}{2}</math> &nbsp; and go towards &nbsp; <math style="vertical-align: -13px">\frac{\pi}{2},</math>
 |-
-|we see that &nbsp;<math style="vertical-align: -5px">\tan(x)</math> &nbsp; goes to &nbsp;<math style="vertical-align: -2px">+\infty.</math>
+|we see that &nbsp;<math style="vertical-align: -5px">\tan(x)</math> &nbsp; goes to &nbsp;<math style="vertical-align: -2px">\infty.</math>
 |-
 |Therefore,
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x)=+\infty.</math>
+|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x)=\infty.</math>
 |}
@@ Line 117: / Line 117: @@
 |&nbsp; &nbsp; '''(b)''' &nbsp; &nbsp; <math>\frac{3}{7}</math>
 |-
-|&nbsp; &nbsp; '''(c)''' &nbsp; &nbsp; <math>+\infty</math>
+|&nbsp; &nbsp; '''(c)''' &nbsp; &nbsp; <math>\infty</math>
 |}
 [[009A_Sample_Midterm_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 21: / Line 21: @@
 !Step 1: &nbsp;
 |-
-||We begin by noticing that we plug in &nbsp;<math style="vertical-align: 0px">x=2</math>&nbsp; into
+|We begin by noticing that we plug in &nbsp;<math style="vertical-align: 0px">x=2</math>&nbsp; into
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{\sqrt{x^2+12}-4}{x-2},</math>

@@ Line 21: / Line 21: @@
 !Step 1: &nbsp;
 |-
-||We begin by noticing that we plug in <math style="vertical-align: 0px">x=2</math> into
+||We begin by noticing that we plug in &nbsp;<math style="vertical-align: 0px">x=2</math>&nbsp; into
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{\sqrt{x^2+12}-4}{x-2},</math>
@@ Line 88: / Line 88: @@
 !Step 1: &nbsp;
 |-
-|We begin by looking at the graph of <math style="vertical-align: -5px">y=\tan(x),</math>
+|We begin by looking at the graph of &nbsp;<math style="vertical-align: -5px">y=\tan(x),</math>
 |-
 |which is displayed below.
@@ Line 98: / Line 98: @@
 !Step 2: &nbsp;
 |-
-|We are taking a left hand limit. So, we approach <math style="vertical-align: -13px">x=\frac{\pi}{2}</math> &nbsp; from the left.
+|We are taking a left hand limit. So, we approach &nbsp;<math style="vertical-align: -13px">x=\frac{\pi}{2}</math> &nbsp; from the left.
 |-
-|If we look at the graph from the left of <math style="vertical-align: -13px">x=\frac{\pi}{2}</math> &nbsp; and go towards &nbsp; <math style="vertical-align: -13px">\frac{\pi}{2},</math>
+|If we look at the graph from the left of &nbsp;<math style="vertical-align: -13px">x=\frac{\pi}{2}</math> &nbsp; and go towards &nbsp; <math style="vertical-align: -13px">\frac{\pi}{2},</math>
 |-
-|we see that <math style="vertical-align: -5px">\tan(x)</math> &nbsp; goes to <math style="vertical-align: -2px">+\infty.</math>
+|we see that &nbsp;<math style="vertical-align: -5px">\tan(x)</math> &nbsp; goes to &nbsp;<math style="vertical-align: -2px">+\infty.</math>
 |-
 |Therefore,

@@ Line 1: / Line 1: @@
 <span class="exam">Evaluate the following limits.
-<span class="exam">(a) Find <math style="vertical-align: -14px">\lim _{x\rightarrow 2} \frac{\sqrt{x^2+12}-4}{x-2}</math>
+<span class="exam">(a) Find &nbsp;<math style="vertical-align: -14px">\lim _{x\rightarrow 2} \frac{\sqrt{x^2+12}-4}{x-2}</math>
-<span class="exam">(b) Find <math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)} </math>
+<span class="exam">(b) Find &nbsp;<math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\sin(3x)}{\sin(7x)} </math>
-<span class="exam">(c) Evaluate <math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math>
+<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -20px">\lim _{x\rightarrow (\frac{\pi}{2})^-} \tan(x) </math>