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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