Evaluating limits of rational functions

[artwill872]

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?

 

[Dick]

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.

 

[HallsofIvy]

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 $+\infty$ (which, since $\infty$ is not a real number is the same as saying the limit does not exist.

(Comparing highest powers is useful when x goes to $\pm\infty$.)