What is the Method for Determining Infinity Limits in a Rational Function?

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SUMMARY

The method for determining limits at infinity for rational functions involves analyzing the degrees of the numerator and denominator. In the example provided, lim (5x^3 + 27)/(20x^2 + 10x + 9) as x approaches negative infinity results in negative infinity. The correct approach is to divide the numerator and denominator by the highest power of x in the denominator, which simplifies the expression and reveals the behavior of the function as x approaches infinity or negative infinity. The conclusion is that since the degree of the numerator (3) is greater than that of the denominator (2), the limit does not exist in a traditional sense and trends towards negative infinity.

PREREQUISITES
  • Understanding of rational functions and their properties
  • Knowledge of limits and continuity in calculus
  • Familiarity with polynomial degrees and their implications
  • Ability to perform algebraic manipulation, such as dividing by the highest power of x
NEXT STEPS
  • Study the concept of limits at infinity in calculus
  • Learn how to apply L'Hôpital's Rule for indeterminate forms
  • Explore polynomial long division for rational functions
  • Investigate the behavior of functions as they approach vertical and horizontal asymptotes
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Students studying calculus, mathematics educators, and anyone seeking to master the evaluation of limits in rational functions.

camboguy
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ok I am confused when x->negative infinity or positive infinity.

for example

lim (5x^3+27)/(20x^2 + 10x + 9)
x-> negative infinty

heres what i think, i want to know if i have the right idea or not.

- so since the top exponent is larger then the denominator the lim DNE and so i plugged in a negative value to test if it is negative or positive infinity so i put in -1 just to test, the thing is i only plunged it into the (5x^3)/(20x^2); I am thinking this is what i do, and then i get a negative value so i am assuming it is negative infinity. that is what the answer is so posed to be, but did i do it how its so posed to be done?
 
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Divide top and bottom by x^2 and get rid of stuff that tends to zero.
 

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