Difference between revisions of "009A Sample Midterm 1, Problem 1"
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!Foundations: | !Foundations: | ||
|- | |- | ||
| − | | | + | | '''1.''' Linearity rules of limits |
|- | |- | ||
| − | | | + | | '''2.''' Limit sin(x)/x |
| − | |||
|- | |- | ||
| − | | | + | |'''3.''' Left and right hand limits |
| − | |||
|} | |} | ||
| Line 24: | Line 22: | ||
!Step 1: | !Step 1: | ||
|- | |- | ||
| − | | | + | |Since <math>\lim_{x\rightarrow 2} x =2\ne 0,</math> |
| + | |- | ||
| + | |we have | ||
|- | |- | ||
| − | | | + | | <math>\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}</math> | ||
|} | |} | ||
| Line 32: | Line 38: | ||
!Step 2: | !Step 2: | ||
|- | |- | ||
| − | | | + | |If we multiply both sides of the last equation by <math>2,</math> we get |
| + | |- | ||
| + | | <math>10=\lim_{x\rightarrow 2} (4-g(x)).</math> | ||
| + | |- | ||
| + | |Now, using linearity properties of limits, we have | ||
| + | |- | ||
| + | | <math>\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}</math> | ||
| + | |} | ||
| + | |||
| + | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Step 3: | ||
| + | |- | ||
| + | |Solving for <math>\lim_{x\rightarrow 2} g(x)</math> in the last equation, | ||
| + | |- | ||
| + | |we get | ||
|- | |- | ||
| | | | ||
| + | <math> \lim_{x\rightarrow 2} g(x)=-6.</math> | ||
|} | |} | ||
| Line 91: | Line 116: | ||
!Final Answer: | !Final Answer: | ||
|- | |- | ||
| − | |'''(a)''' | + | | '''(a)''' <math> \lim_{x\rightarrow 2} g(x)=-6</math> |
|- | |- | ||
|'''(b)''' | |'''(b)''' | ||
Revision as of 08:14, 16 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
| Foundations: |
|---|
| 1. Linearity rules of limits |
| 2. Limit sin(x)/x |
| 3. Left and right hand limits |
Solution:
(a)
| Step 1: |
|---|
| Since |
| 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: |
|---|
| Step 2: |
|---|
(c)
| Step 1: |
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| 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 2} g(x)=-6} |
| (b) |
| (c) |