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Trigonometry Examples
Step 1
Find where the expression is undefined.
Step 2
Consider the rational function where is the degree of the numerator and is the degree of the denominator.
1. If , then the x-axis, , is the horizontal asymptote.
2. If , then the horizontal asymptote is the line .
3. If , then there is no horizontal asymptote (there is an oblique asymptote).
Step 3
Find and .
Step 4
Since , there is no horizontal asymptote.
No Horizontal Asymptotes
Step 5
Step 5.1
Combine.
Step 5.1.1
To write as a fraction with a common denominator, multiply by .
Step 5.1.2
Combine the numerators over the common denominator.
Step 5.1.3
Multiply by by adding the exponents.
Step 5.1.3.1
Use the power rule to combine exponents.
Step 5.1.3.2
Add and .
Step 5.1.4
To write as a fraction with a common denominator, multiply by .
Step 5.1.5
To write as a fraction with a common denominator, multiply by .
Step 5.1.6
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 5.1.6.1
Multiply by .
Step 5.1.6.2
Multiply by .
Step 5.1.6.3
Reorder the factors of .
Step 5.1.7
Combine the numerators over the common denominator.
Step 5.1.8
Simplify the numerator.
Step 5.1.8.1
Apply the distributive property.
Step 5.1.8.2
Move to the left of .
Step 5.1.8.3
Multiply by .
Step 5.1.8.4
Reorder terms.
Step 5.1.9
Simplify.
Step 5.2
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
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Step 5.3
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 5.4
Multiply the new quotient term by the divisor.
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Step 5.5
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 5.6
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 5.7
Pull the next terms from the original dividend down into the current dividend.
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Step 5.8
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 5.9
Multiply the new quotient term by the divisor.
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Step 5.10
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 5.11
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 5.12
The final answer is the quotient plus the remainder over the divisor.
Step 5.13
The oblique asymptote is the polynomial portion of the long division result.
Step 6
This is the set of all asymptotes.
Vertical Asymptotes:
No Horizontal Asymptotes
Oblique Asymptotes:
Step 7