Limits? Maybe. Need help urgently.

Sep 2008
2
0
I've been working on this problem for a while and it's got me stumped.

http://img59.imageshack.us/img59/1536/calc1ho7.png

I thought I was doing it right, until in the third column I got Undefined for where x = 2 and f(x) = 4. I don't need the entire column filled it, just a couple values maybe.

What I've been doing is, for example, for the first pairing (1, 1) I plugged in values for the equation winding up with;

(1-4)/(1-2)
-3/-1
3

I got the 4 by going to the x value on the table and looking at the f(x) that matched up with it which was 4. However... I was worried I was making a mistake when I got undefined for (2,4) and then I was positive I'd made a mistake when I tried to find the speed of the car at 2... where it was undefined. We're currently learning about limits and continuity... and I'm having trouble seeing how this ties in as well.

Edit// Probably should have put this in urgent homework help. Sorry.
 
Jun 2008
792
424
You did not make a mistake when you said it was undefined. Since there is a division by zero when we plug in x = 2, we find E(x) for values of x near 2 from the left side and the right side. Find E(x) for values such as 1.9, 1.99, etc.. and 2.1, 2.01, etc.. You can see that it converges to some value. This is the \(\displaystyle \lim_{x \to 2} E(x)\).
 
Sep 2008
2
0
Okay, well it's nice to know I didn't make a mistake--and I can understand the , but does this simply mean that I cannot find the speed of the car at 2 (my teacher also referred to it as f'(2) which confused us all and was the result of his changing it) and my professor is merely trying to trick me into thinking I did something wrong?
 
Jun 2008
792
424
Are you just beginning Calculus? You should understand the concept of limits and derivatives to do this. First of all, in a limit, when we want to find what value is f(x) converging to when it approaches a certain x, we try to get rid of the discontinuity at x = 2 by manipulating the function (factoring, conjugate, LCD, etc..) into a function where plugging in 2 does not cause a division by 0. Since the function here is not given, we simply plug in numbers near 2 from both sides and estimate the limiting value.

The limit in question here is similar to the definition of the derivative, which is the slope of the tangent line at a given point a. The definition can be either:

\(\displaystyle f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\)

\(\displaystyle f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}\)

We indicate that this is the first derivative of a function by writing f'(x) (f prime of x). Others may use Leibniz notation or Newtons notation, but Lagrange notation (this one) is easier for beginners (in my opinion).

f'(2) gives the slope of the tangent line at x = 2.