Homework Help: Continuity and Limits

1. Sep 6, 2010

Frankie715

1. The problem statement, all variables and given/known data

1)

2)

2. Relevant equations

3. The attempt at a solution

1) I have done plenty of these, but this one is stumping me. I tried plugged in 0 approach for h and I got 11-11=0. With h on the bottom as 0. I know this isn't the right answer. I also know the limit does exist. If anyone could help me with HOW you come to the conclusion I would greatly appreciate it.

2) I have pasted the problem with my attempts at an answer. I am having a little bit of trouble here trying to determine exactly what intervals are continuous and why.

2. Sep 6, 2010

DrummingAtom

Try using an Algebraic Conjugate.

3. Sep 6, 2010

eumyang

1) Rationalize the NUMERATOR.

2) By my count, there are two more intervals where g is continuous. What is happening at x = 4?

Last edited by a moderator: Sep 15, 2010
4. Sep 6, 2010

Frankie715

1) I got 11 out of the square root of 121 which cancels with the other -11. Confused about the remaining h.

2) I was confused at x = 4. Are my current 3 answers correct?

5. Sep 6, 2010

eumyang

1) No, you can't do that. You have a square root of a SUM, and you can't split into a sum of two square roots:
$$\sqrt{121 + h} \ne 11 + \sqrt{h}$$

You rationalize the numerator by multiplying top and bottom by the numerator's conjugate. For example, if the numerator of a fraction is
$$a + \sqrt{b}$$,
then you multiply top and bottom by
$$a - \sqrt{b}$$,
because in doing so, you make the numerator a rational number (hence, rationalizing).
$$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$$.

Last edited by a moderator: Sep 15, 2010
6. Sep 6, 2010

Frankie715

Ok, let me figure this out.

So I multiply the numerator and the denominator by

sqrt (121+h)+11?

Last edited: Sep 6, 2010
7. Sep 7, 2010

eumyang

Yes. Keep going. What happens next?

Last edited by a moderator: Sep 15, 2010
8. Sep 7, 2010

n.karthick

If you substitute h=0 you get 0/0 ie., an indeterminate form. So you can use L' Hospital's rule for this problem.
i.e., differentiate numerator and denominator separately and find the limit of their ratios.

9. Sep 7, 2010

Frankie715

I got h on the top. (sqrt (h + 121)) h + 11h on bottom. Is that correct?

Last edited by a moderator: Sep 15, 2010
10. Sep 8, 2010

Bohrok

That's right. Now notice that each term in the denominator has a factor of h. After you factor out the h, what can you do next?

11. Sep 8, 2010

eumyang

This may not be helpful if the OP is beginning Calc. I at a uni. in the U.S. He/she may have just started to study limits and so he/she would not know what L'HÃ´pital's rule (not L'Hospital ) is.

Last edited by a moderator: Sep 15, 2010