L’Hôpital’s Rule

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L’Hôpital’s Rule

From “Applying L’Hôpital’s Rule” in section 4.8 of our textbook, and the L’Hôpital’s Rule discussion.

An Example

Find lim_x→0 (e^x-1)/x

At the start, notice that this isn’t a limit we could find a clear value for using the methods we already know: you could split the fraction into a difference of 2 fractions, use limit laws on that difference, etc., but even so you’d end up with something indeterminate, like ±∞ / ±∞. So this is one of many examples of the kind of limit that we couldn’t do before, but now can with L’Hôpital’s Rule.

L’Hôpital’s Rule says that a limit of the form f(x / g(x) is equal to the limit of the derivatives, i.e., f′(x) / g′(x), as long as the original limit becomes one of the indeterminate forms 0/0 or ∞/∞.

So start solving this problem by checking that it does indeed have the required form:

Seeing that it does, take derivatives and then try taking the limit again:

Polynomials vs Exponentials and Logarithms

Find lim_x→∞ x/e^x

This limit gives rise to an indeterminate form to which L’Hôpital’s Rule applies, so take the limit of the derivatives. Note that we can find the limit of 1 / e^x by informally reasoning that as x gets larger, so will e^x, without ever reaching any bound. So the reciprocal of e^x will get ever closer to 0, without bound.

Find lim_x→∞ x²/e^x

This is interesting for 2 reasons. First, you can use L’Hôpital’s Rule twice to find this limit — multiple uses of L’Hôpital’s Rule are often a good way to find a limit:

Second, there seems to be something of a pattern here, that powers of x over e^x go to 0 as x goes to infinity. This is always true, since for any finite power you can use L’Hôpital’s Rule repeatedly, like we did for x², until you get a limit of a constant over e^x, which will be 0. Since you can think of the fraction xⁿ / e^x as measuring how much bigger xⁿ is than e^x, you can interpret this result as saying that as x gets larger and larger, all polynomials, no matter how big, become vanishingly small compared to an exponential function.

Find lim_x→∞ x/ln x.

This is also a candidate for L’Hôpital’s Rule:

Finally, find lim_x→∞ √x/ln x

Once again L’Hôpital’s Rule applies, although the derivative is a little harder to find, and after taking derivatives you need to realize that x / √x simplifies to √x:

This pair of limits illustrates a rule analogous to the one about polynomials and exponential functions: any power of x, no matter how small, grows faster than any logarithm of x.

Sometimes you can put functions that seem to have indeterminate limits to which L’Hôpital’s Rule doesn’t apply into forms to which it does. For example, consider the limit as x approaches 0 (from the right) of x cscx. At first glance, this seems to have the indeterminate form 0 × ∞:

But then recall that cosecant is the reciprocal of sine, and use that to rewrite x cscx:

Now the indeterminate form is 0/0, and L’Hôpital’s Rule does apply!

Next

Start studying (definite) integrals, i.e., areas under graphs.

Very early in this course we had a preview of calculus that suggested that areas under curves can be approximated as the sum of the areas of lots of small rectangles tucked under the curve. As the rectangles get smaller this approximation gets better.

So the first thing you need in order to work with integrals is an understanding of sums.

Please read “Sigma (Summation) Notation” in section 5.1 of the textbook by class time tomorrow.

Please also participate in this discussion of summations.

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