Find the Asymptotes (x^2+5x-3)/(4x-1)


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
Find the oblique asymptote using polynomial division.

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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Divide the highest order term in the dividend by the highest order term in divisor .
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Multiply the new quotient term by the divisor.
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The expression needs to be subtracted from the dividend, so change all the signs in
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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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-+
+
Pull the next terms from the original dividend down into the current dividend.
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-+
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Divide the highest order term in the dividend by the highest order term in divisor .
+
-+-
-+
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Multiply the new quotient term by the divisor.
+
-+-
-+
+-
+-
The expression needs to be subtracted from the dividend, so change all the signs in
+
-+-
-+
+-
-+
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
+
-+-
-+
+-
-+
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The final answer is the quotient plus the remainder over the divisor.
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


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