Difference between revisions of "Math 22 Asymptotes"
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==Vertical Asymptotes and Infinite Limits== | ==Vertical Asymptotes and Infinite Limits== | ||
| + | This page is under construction | ||
If <math>f(x)</math> approaches infinity (or negative infinity) as <math>x</math> approaches <math>c</math> | If <math>f(x)</math> approaches infinity (or negative infinity) as <math>x</math> approaches <math>c</math> | ||
from the right or from the left, then the line <math>x=c</math> is a vertical asmptote of the graph of <math>f</math> | from the right or from the left, then the line <math>x=c</math> is a vertical asmptote of the graph of <math>f</math> | ||
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|} | |} | ||
| − | + | ==Definition of Horizontal Asymptote== | |
| + | |||
| + | If <math>f</math> is a function and <math>L_1</math> and <math>L_2</math> are real numbers, then the statements | ||
| + | <math>\lim_{x\to\infty} f(x)=L_1</math> and <math>\lim_{x\to -\infty} f(x)=L_2</math> | ||
| + | denote limits at infinity. The line <math>y=L_1</math> and <math>y=L_2</math> are horizontal asymptotes | ||
| + | of the graph of <math>f</math> | ||
| + | |||
| + | ==Horizontal Asymptotes of Rational Functions== | ||
| + | Let <math>f(x)=\frac{p(x)}{q(x)}</math> be a rational function. | ||
| + | 1. If the degree of the numerator is less than the degree of the denominator, | ||
| + | then is a horizontal asymptote of the graph of (to the left and to the right). | ||
| + | 2. If the degree of the numerator is equal to the degree of the denominator, | ||
| + | then is a horizontal asymptote of the graph of (to the left and to the right), | ||
| + | where and are the leading coefficients of and , respectively. | ||
| + | 3. If the degree of the numerator is greater than the degree of the denominator, | ||
| + | then the graph of has no horizontal asymptote. | ||
[[Math_22| '''Return to Topics Page''']] | [[Math_22| '''Return to Topics Page''']] | ||
'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' | ||
Latest revision as of 08:34, 23 October 2020
Vertical Asymptotes and Infinite Limits
This page is under construction
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 f(x)}
approaches infinity (or negative infinity) 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}
from the right or from the left, then the line 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=c}
is a vertical asmptote of the graph 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}
Example: Find the a vertical Asymptotes as below:
1) 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)=\frac{x+3}{x^2-4}}
| ExpandSolution: |
|---|
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 f(x)=\frac{x^2-x-6}{x^2-9}}
| ExpandSolution: |
|---|
Definition of Horizontal Asymptote
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 f}
is a function 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 L_1}
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 L_2}
are real numbers, then the statements
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\infty} f(x)=L_1}
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 -\infty} f(x)=L_2}
denote limits at infinity. The line 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 y=L_1}
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 y=L_2}
are horizontal asymptotes
of the graph 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}
Horizontal Asymptotes of Rational Functions
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(x)=\frac{p(x)}{q(x)}}
be a rational function.
1. If the degree of the numerator is less than the degree of the denominator,
then is a horizontal asymptote of the graph of (to the left and to the right).
2. If the degree of the numerator is equal to the degree of the denominator,
then is a horizontal asymptote of the graph of (to the left and to the right),
where and are the leading coefficients of and , respectively.
3. If the degree of the numerator is greater than the degree of the denominator,
then the graph of has no horizontal asymptote.
This page were made by Tri Phan