Difference between revisions of "009A Sample Midterm 3, Problem 1"
Jump to navigation
Jump to search
Kayla Murray (talk | contribs) |
Kayla Murray (talk | contribs) |
||
| Line 59: | Line 59: | ||
!Step 1: | !Step 1: | ||
|- | |- | ||
| − | | | + | |First, we write |
| − | |||
| − | |||
| − | |||
| − | |||
|- | |- | ||
| − | | | + | | <math>\begin{array}{rcl} |
| + | \displaystyle{\lim_{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}} & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(4x)}{\cos(4x)} \frac{1}{\sin(6x)}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\lim_{x\rightarrow 0} \frac{4}{6} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\frac{4}{6}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}.} | ||
| + | \end{array}</math> | ||
|} | |} | ||
| Line 71: | Line 73: | ||
!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | | |Now, we have |
| − | |||
| − | |||
| − | |||
| − | | | ||
|- | |- | ||
| | | | ||
| + | <math>\begin{array}{rcl} | ||
| + | \displaystyle{\lim_{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}} & = & \displaystyle{\frac{4}{6}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\frac{4}{6}\bigg(\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{6x}{\sin(6x)}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{1}{\cos(4x)}\bigg)}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\frac{4}{6} (1)(1)(1)}\\ | ||
| + | &&\\ | ||
| + | & = & \displaystyle{\frac{2}{3}.} | ||
| + | \end{array}</math> | ||
|} | |} | ||
| Line 103: | Line 110: | ||
| '''(a)''' <math>\lim_{x\rightarrow 3} f(x)=6</math> | | '''(a)''' <math>\lim_{x\rightarrow 3} f(x)=6</math> | ||
|- | |- | ||
| − | |'''(b)''' | + | | '''(b)''' <math>\frac{2}{3}</math> |
|- | |- | ||
|'''(c)''' | |'''(c)''' | ||
|} | |} | ||
[[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | [[009A_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | ||
Revision as of 13:07, 17 February 2017
Find the following limits:
- a) If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)=2,} find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 3} f(x).}
- b) Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}. }
- c) Evaluate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim _{x\rightarrow \infty} \frac{-2x^3-2x+3}{3x^3+3x^2-3}. }
| Foundations: |
|---|
| 1. Linearity rules of limits |
| 2. lim sin(x)/x |
Solution:
(a)
| Step 1: |
|---|
| First, we have |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{2} & = & \displaystyle{\lim_{x\rightarrow 3} \bigg(\frac{f(x)}{2x}+1\bigg)}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+\lim_{x\rightarrow 3} 1}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}+1.} \end{array}} |
| Therefore, |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow 3} \frac{f(x)}{2x}=1.} |
| Step 2: |
|---|
| Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow 3} 2x=6\ne 0,} we have |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{1} & = & \displaystyle{\lim_{x\rightarrow 3} \frac{f(x)}{2x}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 3} f(x)}{\lim_{x\rightarrow} 2x}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 3} f(x)}{6}.} \end{array}} |
| Multiplying both sides by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6,} we get |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow 3} f(x)=6.} |
(b)
| Step 1: |
|---|
| First, we write |
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\lim_{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}} & = & \displaystyle{\lim_{x\rightarrow 0} \frac{\sin(4x)}{\cos(4x)} \frac{1}{\sin(6x)}}\\ &&\\ & = & \displaystyle{\lim_{x\rightarrow 0} \frac{4}{6} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}}\\ &&\\ & = & \displaystyle{\frac{4}{6}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}.} \end{array}} |
| Step 2: |
|---|
| Now, we have |
|
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\lim_{x\rightarrow 0} \frac{\tan(4x)}{\sin(6x)}} & = & \displaystyle{\frac{4}{6}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\frac{6x}{\sin(6x)}\frac{1}{\cos(4x)}}\\ &&\\ & = & \displaystyle{\frac{4}{6}\bigg(\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{6x}{\sin(6x)}\bigg)\bigg(\lim_{x\rightarrow 0} \frac{1}{\cos(4x)}\bigg)}\\ &&\\ & = & \displaystyle{\frac{4}{6} (1)(1)(1)}\\ &&\\ & = & \displaystyle{\frac{2}{3}.} \end{array}} |
(c)
| Step 1: |
|---|
| Step 2: |
|---|
| Final Answer: |
|---|
| (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow 3} f(x)=6} |
| (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2}{3}} |
| (c) |