# 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'Hôpital's rule (not L'Hospital ) is.

Last edited by a moderator: Sep 15, 2010