009A Sample Midterm 3, Problem 1 - Revision history

Kayla Murray at 01:51, 7 November 2017

2017-11-07T01:51:21Z

2017-11-03T17:53:44Z

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@@ Line 9: / Line 9: @@
 <hr>
-(insert picture of handwritten solution)
+[[009A Sample Midterm 3, Problem 1 Detailed Solution|'''<u>Detailed Solution with Background Information</u>''']]
-[[009A Sample Midterm 3, Problem 1 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']]
+[[File:9ASM3P1.jpg|600px|thumb|center]]
 [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

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 <span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -16px">\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. </math>
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+(insert picture of handwritten solution)
 [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

@@ Line 15: / Line 15: @@
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow a} \frac{f(x)}{g(x)}=\frac{\displaystyle{\lim_{x\rightarrow a} f(x)}}{\displaystyle{\lim_{x\rightarrow a} g(x)}}.</math>
 |-
-|'''2.''' &nbsp;<math style="vertical-align: -15px">\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math>
+|'''2.''' Recall
 |}

@@ Line 127: / Line 127: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>\lim_{x\rightarrow 3} f(x)=6</math>
+|&nbsp; &nbsp; '''(a)''' &nbsp; &nbsp; <math>6</math>
 |-
 |&nbsp; &nbsp; '''(b)'''  &nbsp; &nbsp; <math>\frac{2}{3}</math>

@@ Line 11: / Line 11: @@
 !Foundations: &nbsp;
 |-
-|'''1.''' If <math style="vertical-align: -13px">\lim_{x\rightarrow a} g(x)\neq 0,</math> we have
+|'''1.''' If &nbsp;<math style="vertical-align: -13px">\lim_{x\rightarrow a} g(x)\neq 0,</math>&nbsp; we have
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow a} \frac{f(x)}{g(x)}=\frac{\displaystyle{\lim_{x\rightarrow a} f(x)}}{\displaystyle{\lim_{x\rightarrow a} g(x)}}.</math>
 |-
-|'''2.''' <math style="vertical-align: -15px">\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math>
+|'''2.''' &nbsp;<math style="vertical-align: -15px">\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math>
 |}
@@ Line 43: / Line 43: @@
 !Step 2: &nbsp;
 |-
-|Since <math style="vertical-align: -13px">\lim_{x\rightarrow 3} 2x=6\ne 0,</math> we have
+|Since &nbsp;<math style="vertical-align: -13px">\lim_{x\rightarrow 3} 2x=6\ne 0,</math>&nbsp; we have
 |-
+|
@@ Line 54: / Line 54: @@
 \end{array}</math>
 |-
-|Multiplying both sides by <math style="vertical-align: -5px">6,</math> we get
+|Multiplying both sides by &nbsp;<math style="vertical-align: -5px">6,</math>&nbsp; we get
 |-
 |&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow 3} f(x)=6.</math>

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 <span class="exam"> Find the following limits:
-<span class="exam">(a) If <math style="vertical-align: -16px">\lim _{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)=2,</math> find <math style="vertical-align: -13px">\lim _{x\rightarrow 3} f(x).</math>
+<span class="exam">(a) If &nbsp;<math style="vertical-align: -16px">\lim _{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)=2,</math>&nbsp; find &nbsp;<math style="vertical-align: -13px">\lim _{x\rightarrow 3} f(x).</math>
-<span class="exam">(b) Find <math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}. </math>
+<span class="exam">(b) Find &nbsp;<math style="vertical-align: -19px">\lim _{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}. </math>
-<span class="exam">(c) Evaluate <math style="vertical-align: -16px">\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. </math>
+<span class="exam">(c) Evaluate &nbsp;<math style="vertical-align: -16px">\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. </math>

@@ Line 1: / Line 1: @@
 <span class="exam">Find the following limits:
-<span class="exam">a) If <math>\lim _{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)=2,</math> find <math>\lim _{x\rightarrow 3} f(x).</math>
+<span class="exam">(a) If <math>\lim _{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)=2,</math> find <math>\lim _{x\rightarrow 3} f(x).</math>
-<span class="exam">b) Find <math>\lim _{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}. </math>
+<span class="exam">(b) Find <math>\lim _{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}. </math>
-<span class="exam">c) Evaluate <math>\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. </math>
+<span class="exam">(c) Evaluate <math>\lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. </math>