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Evaluating limits of rational functions

  • Thread starter artwill872
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  • #1

Homework Statement


Why does the limit as x approaches 0 of
x^2 + 5 / 3x go to infinity (with 0 as an essential disc.) but without the +5, the function goes to 0?


Homework Equations





The Attempt at a Solution


I tried approaching evaluating the limit of the function by comparing the exponents of the numerator and denominator and that seems to work if the +5 isn't there. What is the explanation for this difference?
 

Answers and Replies

  • #2
Dick
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You mean (x^2+5)/(3x), right? Parentheses help. Without the +5, you've got x^2/(3x). That's simplifies to x/3, right? So it approaches 0. With the +5 the numerator approaches 5 and the denominator approaches 0, so the quotient is infinity. Counting powers doesn't really help here.
 
  • #3
HallsofIvy
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A little more precisely, for x very, very close to 0, the numerator is close to 5 and the denominator is close to 0. That will be a very, very, large number. As x gets closer to 0, the numerator stays close to 5 while the denominator gets closer to 0. That is, the limit is [itex]+\infty[/itex] (which, since [itex]\infty[/itex] is not a real number is the same as saying the limit does not exist.

(Comparing highest powers is useful when x goes to [itex]\pm\infty[/itex].)
 

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