Math 22 Asymptotes - Revision history

Tphan046: /* Horizontal Asymptotes of Rational Functions */

2020-10-23T16:34:36Z

Horizontal Asymptotes of Rational Functions

Tphan046 at 14:26, 10 August 2020

2020-08-10T14:26:35Z

Tphan046 at 14:21, 10 August 2020

2020-08-10T14:21:15Z

Tphan046 at 16:05, 4 August 2020

2020-08-04T16:05:23Z

Tphan046: /* Definition of Horizontal Asymptote */

2020-08-04T16:05:08Z

Definition of Horizontal Asymptote

Tphan046 at 15:27, 4 August 2020

2020-08-04T15:27:21Z

Tphan046: /* Vertical Asymptotes and Infinite Limits */

2020-08-04T15:10:32Z

Vertical Asymptotes and Infinite Limits

Tphan046: /* Vertical Asymptotes and Infinite Limits */

2020-08-04T15:10:19Z

Vertical Asymptotes and Infinite Limits

Tphan046: /* Vertical Asymptotes and Infinite Limits */

2020-08-04T15:09:36Z

Vertical Asymptotes and Infinite Limits

Tphan046: /* Vertical Asymptotes and Infinite Limits */

2020-08-04T15:07:10Z

Vertical Asymptotes and Infinite Limits

@@ Line 39: / Line 39: @@
 . 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).
 . 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.

@@ Line 44: / Line 44: @@
 . 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''']]
 '''This page were made by [[Contributors|Tri Phan]]'''

@@ Line 35: / Line 35: @@
    of the graph of <math>f</math>
+==Horizontal Asymptotes of Rational Functions==
 [[Math_22| '''Return to Topics Page''']]
 '''This page were made by [[Contributors|Tri Phan]]'''

@@ Line 1: / Line 1: @@
 ==Vertical Asymptotes and Infinite Limits==
    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>

@@ Line 28: / Line 28: @@
 ==Definition of Horizontal Asymptote==
-This page is under construction
    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>

@@ Line 2: / Line 2: @@
    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
-'''Example''': Find the limit below:
+'''Example''': Find the a vertical Asymptotes as below:
-'''1)''' <math>\lim_{x\to 3^-}\frac{-2}{x-3}</math>
+'''1)''' <math>f(x)=\frac{x+3}{x^2-4}</math>
 {| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
 |-
-|'''Important''': <math>\lim \frac{\text{constant}}{0}</math> is either <math>\infty</math> or <math>-\infty</math>
+|Notice <math>f(x)\frac{x+3}{x^2-4}=\frac{x+3}{(x-2)(x+2)}</math>
 |-
-|Notice <math>x\to 3^-</math>, so <math> x<3 </math>, then <math> x-3<0</math>, hence the denominator will be "negative".
+|Let the denominator equals to zero, ie: <math>(x-2)(x+2)=0</math>, hence <math>x=-2</math> or <math>x=2</math>
 |-
-|Therefore, <math>\lim_{x\to 3^-}\frac{-2}{x-3}=\frac{\text{negative}}{\text{negative}}=\infty</math>
+|Therefore, <math>f(x)</math> has vertical asymptotes at <math>x=2</math> and <math>x=-2</math>
 |}
-'''2)''' <math>\lim_{x\to 2^+}\frac{-5}{x-2}</math>
+'''2)''' <math>f(x)=\frac{x^2-x-6}{x^2-9}</math>
 {| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
 |-
-|Notice <math>x\to 2^+</math>, so <math> x>2 </math>, then <math> x-2>0</math>, hence the denominator will be "positive".
+|Notice <math>f(x)\frac{x^2-x-6}{x^2-9}=\frac{(x-3)(x+2)}{(x-3)(x+3)}=\frac{x+2}{x+3}</math>
 |-
-|Therefore, <math>\lim_{x\to 2^+}\frac{-5}{x-2}=\frac{\text{negative}}{\text{positive}}=-\infty</math>
+|Let the denominator equals to zero, ie: <math>(x+3)=0</math>, hence <math>x=-3</math>
 |}
 This page is under construction