What is the Limit of a Rational Function at a Removable Singularity?

[davie08]

Homework Statement

i think I remember the question to be this:

lim (y-3)/(y-3)(y+3)
y→3

it was the first limit question and the easiest but I am not sure if I did it right.

Homework Equations

The Attempt at a Solution

so what I did was I canceled the y-3's out to get:

1/(y+3) and this is what I was unsure about is that the first step and after this I got:


1/(3+3) = 1/6

 

[DivisionByZro]

You're correct:

$$<br /> \lim_{y \to \ 3} \frac{y-3}{(y-3)(y+3)} = \lim_{y \to \ 3 } \frac{1}{y+3} = \frac{1}{6}<br />$$

 

[spamiam]

You have the correct answer, but it's probably a good idea to try to justify your last step. When is it true that $\lim_{y \to a} f(y) = f(a)$? What can you say about the continuity of rational functions?

 

[HallsofIvy]

A crucial, and often overlooked property of limits is this: if f(x)= g(x) for all x except x= a, then $\limit_{x\to a} f(x)= \limit_{x\to a}g(x)$.

For this problem, you can correctly say that
$$\frac{y-3}{(y-3)(y+3)}= \frac{1}{y+3}$$
for all x except x= 3 and so the limits are the same.

Actually, "what can you say about the continuity of rational functions?" is NOT a good question here because continuity also involves the value of the function as well as the limit. 1/(x+3) is continuous at x= 3, so its limit is 1/(3+3)= 1/6 while (x-3)/(x-3)(x+3) is NOT continuous at x= 3.

 

[spamiam]

1/(x+3) is continuous at x= 3, so its limit is 1/(3+3)= 1/6

This statement is true exactly because rational functions are continuous anywhere their denominator isn't zero, which is why I asked the question. I guess I should have mentioned the fact that there is a removable singularity at x=3, but the OP seemed to handle it appropriately.