Problem when rationalizing a function in limits

[Sirus Glaceon]

So, when we come up with some limits like

[FONT=&quot](x+1)/(√(x+5)-2), we would rationalize it and obtain the answer. However I am wondering if this will change the value of the function. We rationalize a function based on the notion that (a/a) equals to 1, which will not affect the value of the function. But take f(x)=5x-1 as an example. This function is continuous. But if we multiply f(x) with (x^2-4)/(x^2-4), then there is an undefined value(a hole) at x=2, which means that the value of the function has changed. Can anyone explain this to me? Very much thanks![/FONT]​

 

[HallsofIvy]

Actually, the example you give has a hole at both x= 2 and x= -2. Yes, that does change the value of the function. You will sometimes see people write, incorrectly, that \(\displaystyle \frac{(x- 2)(x- 1)}{x- 2}= x- 1\). Those are equal everywhere except at x= 2. But that kind of calculation is typically done in computing limits- and the limits of the two function, as x goes to 2, are the same, 1.

Of course, multiplying a function by something (other than 1) changes that function! And, again, \(\displaystyle \frac{x^2- 4}{x^2- 4}\) is equal to 1 everywhere except at x= 2 and x= -2.

 

[Otis]

Good question!

Functions have a domain.

Issues that arise from values that are not in the domain may generally be dealt with by appending a domain statement.

And, always remember, in a limit statement, the variable is only approaching some fixed value; the variable never takes on that value. :cool: