Difference between revisions of "Math 22 Limits"
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Note: Many times the limit of <math>f(x)</math> as <math>x</math> approaches <math>c</math> is simply <math>f(c)</math>, so limit can be evaluate by '''direct substitution''' as <math>\lim_{x\to c} f(x)=f(c)</math> | Note: Many times the limit of <math>f(x)</math> as <math>x</math> approaches <math>c</math> is simply <math>f(c)</math>, so limit can be evaluate by '''direct substitution''' as <math>\lim_{x\to c} f(x)=f(c)</math> | ||
| − | == | + | ==Properties of Limits== |
| − | + | Let <math>b</math> and <math>c</math> be real numbers, let <math>n</math> be a positive integer, and let <math>f</math> and <math>g</math> be functions with the following limits | |
| − | + | <math>\lim_{x\to c} f(x)=L</math> and <math>\lim_{x\to c} g(x)=K</math>. | |
| − | + | Then | |
| − | + | 1. Scalar multiple: <math>\lim_{x\to c} [bf(x)]=bL</math> | |
| − | + | ||
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This page is under constuction | This page is under constuction | ||
'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' | ||
Revision as of 20:31, 13 July 2020
The Limit of a Function
Definition of the Limit of a Function If becomes arbitrarily close to a single number as approaches from either side, then which is read as "the limit of as approaches is 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 L}
Note: Many times the limit 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 f(x)} as 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} approaches 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 c} is simply 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 f(c)} , so limit can be evaluate by direct substitution as 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\to c} f(x)=f(c)}
Properties of Limits
Let 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 b} 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 c} be real numbers, let 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 n} be a positive integer, and let 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 f} 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 g} be functions with the following limits 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\to c} f(x)=L} 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 \lim_{x\to c} g(x)=K} . Then 1. Scalar multiple: 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\to c} [bf(x)]=bL}
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This page were made by Tri Phan