Difference between revisions of "009A Sample Midterm 1, Problem 1"

From Grad Wiki
Jump to navigation Jump to search
Line 11: Line 11:
 
!Foundations:    
 
!Foundations:    
 
|-
 
|-
−
| '''1.''' If <math style="vertical-align: -12px">\lim_{x\rightarrow a} g(x)\neq 0,</math> we have
+
| '''1.''' If &nbsp;<math style="vertical-align: -12px">\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>
 
|&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: -14px">\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math>
+
| '''2.''' &nbsp;<math style="vertical-align: -14px">\lim_{x\rightarrow 0} \frac{\sin x}{x}=1</math>
 
|}
 
|}
  
Line 25: Line 25:
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|Since <math style="vertical-align: -12px">\lim_{x\rightarrow 2} x =2\ne 0,</math>
+
|Since &nbsp;<math style="vertical-align: -12px">\lim_{x\rightarrow 2} x =2\ne 0,</math>
 
|-
 
|-
 
|we have
 
|we have
Line 41: Line 41:
 
!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
−
|If we multiply both sides of the last equation by <math>2,</math> we get
+
|If we multiply both sides of the last equation by &nbsp;<math>2,</math>&nbsp; we get
 
|-
 
|-
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>10=\lim_{x\rightarrow 2} (4-g(x)).</math>
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>10=\lim_{x\rightarrow 2} (4-g(x)).</math>
Line 57: Line 57:
 
!Step 3: &nbsp;
 
!Step 3: &nbsp;
 
|-
 
|-
−
|Solving for <math style="vertical-align: -12px">\lim_{x\rightarrow 2} g(x)</math> in the last equation,
+
|Solving for &nbsp;<math style="vertical-align: -12px">\lim_{x\rightarrow 2} g(x)</math>&nbsp; in the last equation,
 
|-
 
|-
 
|we get
 
|we get
Line 92: Line 92:
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|When we plug in <math style="vertical-align: 0px">-3</math> into &nbsp; <math style="vertical-align: -12px">\frac{x}{x^2-9},</math>
+
|When we plug in &nbsp;<math style="vertical-align: 0px">-3</math>&nbsp; into &nbsp; <math style="vertical-align: -12px">\frac{x}{x^2-9},</math>
 
|-
 
|-
 
|we get &nbsp; <math style="vertical-align: -12px">\frac{-3}{0}.</math>  
 
|we get &nbsp; <math style="vertical-align: -12px">\frac{-3}{0}.</math>  
Line 100: Line 100:
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow -3^+} \frac{x}{x^2-9}</math>  
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow -3^+} \frac{x}{x^2-9}</math>  
 
|-
 
|-
−
|is either equal to <math style="vertical-align: -1px">+\infty</math> or <math style="vertical-align: -1px">-\infty.</math>
+
|is either equal to &nbsp;<math style="vertical-align: -1px">+\infty</math>&nbsp; or &nbsp;<math style="vertical-align: -1px">-\infty.</math>
 
|}
 
|}
  
Line 110: Line 110:
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow -3^+} \frac{x}{x^2-9}=\lim_{x\rightarrow -3^+} \frac{x}{(x-3)(x+3)}.</math>
 
|&nbsp; &nbsp; &nbsp; &nbsp; <math>\lim_{x\rightarrow -3^+} \frac{x}{x^2-9}=\lim_{x\rightarrow -3^+} \frac{x}{(x-3)(x+3)}.</math>
 
|-
 
|-
−
|We are taking a right hand limit. So, we are looking at values of <math style="vertical-align: 0px">x</math>  
+
|We are taking a right hand limit. So, we are looking at values of &nbsp;<math style="vertical-align: 0px">x</math>  
 
|-
 
|-
−
|a little bigger than <math style="vertical-align: 0px">-3.</math> (You can imagine values like <math style="vertical-align: 0px">x=-2.9.</math>)
+
|a little bigger than &nbsp;<math style="vertical-align: 0px">-3.</math>&nbsp; (You can imagine values like &nbsp;<math style="vertical-align: 0px">x=-2.9.</math>&nbsp;)
 
|-
 
|-
 
|For these values, the numerator will be negative.   
 
|For these values, the numerator will be negative.   
 
|-
 
|-
−
|Also, for these values, <math style="vertical-align: 0px">x-3</math> will be negative and <math style="vertical-align: -1px">x+3</math> will be positive.  
+
|Also, for these values, &nbsp;<math style="vertical-align: 0px">x-3</math>&nbsp; will be negative and &nbsp;<math style="vertical-align: -1px">x+3</math>&nbsp; will be positive.  
 
|-
 
|-
 
|Therefore, the denominator will be negative.  
 
|Therefore, the denominator will be negative.  

Revision as of 15:43, 26 February 2017

Find the following limits:

(a) 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 2} g(x),}   provided that  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 2} \bigg[\frac{4-g(x)}{x}\bigg]=5.}

(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{\sin(4x)}{5x} }

(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 -3^+} \frac{x}{x^2-9} }


Foundations:  
1. 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 a} g(x)\neq 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 \lim_{x\rightarrow a} \frac{f(x)}{g(x)}=\frac{\displaystyle{\lim_{x\rightarrow a} f(x)}}{\displaystyle{\lim_{x\rightarrow a} g(x)}}.}
2.  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{\sin x}{x}=1}


Solution:

(a)

Step 1:  
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 2} x =2\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{5} & = & \displaystyle{\lim _{x\rightarrow 2} \bigg[\frac{4-g(x)}{x}\bigg]}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 2} (4-g(x))}{\lim_{x\rightarrow 2} x}}\\ &&\\ & = & \displaystyle{\frac{\lim_{x\rightarrow 2} (4-g(x))}{2}.} \end{array}}
Step 2:  
If we multiply both sides of the last equation 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 2,}   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 10=\lim_{x\rightarrow 2} (4-g(x)).}
Now, using linearity properties of limits, 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{10} & = & \displaystyle{\lim_{x\rightarrow 2} 4 -\lim_{x\rightarrow 2}g(x)}\\ &&\\ & = & \displaystyle{4-\lim_{x\rightarrow 2} g(x).}\\ \end{array}}
Step 3:  
Solving for  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 2} g(x)}   in the last equation,
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 2} g(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 \lim_{x\rightarrow 0} \frac{\sin(4x)}{5x}=\lim_{x\rightarrow 0} \frac{4}{5} \bigg(\frac{\sin(4x)}{4x}\bigg).}
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{\sin(4x)}{5x}} & = & \displaystyle{\frac{4}{5}\lim_{x\rightarrow 0} \frac{\sin(4x)}{4x}}\\ &&\\ & = & \displaystyle{\frac{4}{5}(1)}\\ &&\\ & = & \displaystyle{\frac{4}{5}.} \end{array}}

(c)

Step 1:  
When we plug in  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 -3}   into   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{x}{x^2-9},}
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 \frac{-3}{0}.}
Thus,
        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{x}{x^2-9}}
is either equal to  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 +\infty}   or  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 -\infty.}
Step 2:  
To figure out which one, we factor the denominator to 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^+} \frac{x}{x^2-9}=\lim_{x\rightarrow -3^+} \frac{x}{(x-3)(x+3)}.}
We are taking a right hand limit. So, we are looking at values of  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 x}
a little bigger than  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 -3.}   (You can imagine values like  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 x=-2.9.}  )
For these values, the numerator will be negative.
Also, for these values,  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 x-3}   will be negative and  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 x+3}   will be positive.
Therefore, the denominator will be negative.
Since both the numerator and denominator will be negative (have the same sign),
        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{x}{x^2-9}=+\infty.}


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 2} g(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{4}{5}}
    (c)     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 +\infty}

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009A_Sample_Midterm_1,_Problem_1&oldid=2771"