Have you been given L'Hopital's Rule yet?

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- Sep 26th 2008, 05:31 PM #1

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## i need help

hi everybody, i have some problems from calculus but i can not solve , if someone can help, i will be very glad.

1) lim cos(3x)

x→1 x²+ 2x-1

2) lim sin (x-1)

x→∞ (2+x²)

3) lim sin (x²+1)

x→∞ (x+1)

4) lim (x-1) sin (x+2)

x→ -2 x²-4

5) lim √x +x²

x→-∞ x

- Sep 26th 2008, 05:36 PM #2

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- Sep 26th 2008, 05:45 PM #3

- Sep 26th 2008, 06:01 PM #4

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- Sep 27th 2008, 08:18 AM #5

- Sep 27th 2008, 08:34 AM #6

- Sep 28th 2008, 01:35 AM #7

- Sep 28th 2008, 02:24 AM #8
the is a bounded function, hence if the denominator grows very much, this means shrinks to .

Note that means the denominator ( ) tends to .

**Note**. In general, let be two functions such that and is bounded.

Then, we have .

So, with this note, you can solve 3) and 4) too.

- Sep 28th 2008, 02:29 AM #9

- Sep 28th 2008, 02:34 AM #10

- Sep 28th 2008, 02:37 AM #11
I would have written this way :

for all x.

Now we can apply the limits :

But

So we have the following inequality :

Hence

The "problem" in what you did is that you consider since the beginning the limits. But you have to make the comparison before the limit.

Another thing : in this case, with the inequality, we can say the limit**equals**0. It is no need to say it*tends*to 0.

- Sep 28th 2008, 03:32 AM #12
Why's that?

Surely if I write:

Then that's right since they could never equal each other? Since:

and this has no solutions.

The "problem" in what you did is that you consider since the beginning the limits. But you have to make the comparison before the limit.

Another thing : in this case, with the inequality, we can say the limit equals 0. It is no need to say it tends to 0.

- Sep 28th 2008, 03:37 AM #13
Your two questions are linked. You're working with limits, not functions.

Limits**are**numbers (in this case). The limit equals a number,__it is the__.**value**the function approaches when x goes to infinity

If I write , what is it ?

Yes, you surely have for any x. But their limits are**equal**. This is why you have to consider the week inequalities.

Another point : I don't think there is any need to introduce and . The limits of are well-known

Edit : please don't take the bold and underlined parts badly, I'm just trying to stress out the important points

- Sep 28th 2008, 03:40 AM #14

- Sep 28th 2008, 04:16 AM #15If I write , what is it ?

As this becomes 2x. I write in all my working and ignore the fact that the tiny h exists at the end.

Yes, you surely have for any x. But their limits are equal. This is why you have to consider the week inequalities.

For infinite sequences I also write and not "approximately equal". In the case of infinte sequences is just another limit right?

Another point : I don't think there is any need to introduce and . The limits of are well-known

Edit : please don't take the bold and underlined parts badly, I'm just trying to stress out the important points

What about constant functions? It may also reach it!

Thanks bkarpuz and Moo. I think I get it. When I have these lectures at university everyone is going to think i'm so great!