Find where the expression <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mstyle></math> is undefined.

Consider the rational function <math><mstyle displaystyle="true"><mi>R</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mi>b</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></mfrac></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> is the degree of the numerator and <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> is the degree of the denominator.

1. If <math><mstyle displaystyle="true"><mi>n</mi><mo><</mo><mi>m</mi></mstyle></math> , then the x-axis, <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>0</mn></mstyle></math> , is the horizontal asymptote.

2. If <math><mstyle displaystyle="true"><mi>n</mi><mo>=</mo><mi>m</mi></mstyle></math> , then the horizontal asymptote is the line <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> .

3. If <math><mstyle displaystyle="true"><mi>n</mi><mo>></mo><mi>m</mi></mstyle></math> , then there is no horizontal asymptote (there is an oblique asymptote).

Find <math><mstyle displaystyle="true"><mi>n</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>m</mi></mstyle></math> .

Since <math><mstyle displaystyle="true"><mi>n</mi><mo>=</mo><mi>m</mi></mstyle></math> , the horizontal asymptote is the line <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></mfrac></mstyle></math> where <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mo>-</mo><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mn>1</mn></mstyle></math> .

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

This is the set of all asymptotes.

Vertical Asymptotes: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn></mstyle></math>

Horizontal Asymptotes: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn></mstyle></math>

No Oblique Asymptotes

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Name | two billion thirty-six million nine hundred four thousand seven hundred fifty-five |
---|

- 2036904755 has 16 divisors, whose sum is
**2130036480** - The reverse of 2036904755 is
**5574096302** - Previous prime number is
**263**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 41
- Digital Root 5

Name | seven hundred sixty-eight million six hundred forty-seven thousand seven hundred twenty-two |
---|

- 768647722 has 8 divisors, whose sum is
**1167566400** - The reverse of 768647722 is
**227746867** - Previous prime number is
**79**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 49
- Digital Root 4

Name | one billion five hundred twenty million three hundred thirty-one thousand seven hundred eighteen |
---|

- 1520331718 has 4 divisors, whose sum is
**2280497580** - The reverse of 1520331718 is
**8171330251** - Previous prime number is
**2**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 31
- Digital Root 4