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### Two Somewhat Similar Problems

```
Date: Mon, 9 Jan 1995 09:11:29 -0500 (EST)
From: Secret Someone

Okay, last two (for now):

1.  Evaluate:  lim  Integral from 1 to 1+h of (x^5+8)^.5 dx all over h
h->0
UGH!!@\$@#\$@%#!

2.  If F is differentiable for all x then evaluate

lim  Integral from a to a+h of F'(x) dx all over h
h->0

These two problems seem somewhat similar. (in appearance and in the
fact  that I can't do either one of them)

Thanks for the help,
Burris
```

```
Date: 9 Jan 1995 23:43:20 -0500
From: Dr. Ken
Subject: Re: your mail

Hello there!

Yes, you're right, these two problems are _very_ similar.  Do you know
L'Hopital's rule?  Do you know any other word with two apostrophes in it?
I sure don't.

Anyway, notice that in these limits, the numerators and denominators are
both going to zero.  That means we can use L'Hopital's rule, and take the
derivative with respect to h of the top and the bottom.

So just remember that the derivative of the integral from c to x of f(u) du
is f(x), where c is constant and x is a variable.  That should be enough to
get you through these problems.  Yup, it was a trick.

-Ken "Dr." Math
```
Associated Topics:
High School Calculus

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