L'Hopital's Rule

L'Hopital's Rule

L'Hopital's Rule states that if you have the limits of two functions, and the limit of the product of those functions is equal to the ratio of the limits of each function, then the derivative of that product will equal the derivative of that ratio.

We will conclude our discussion of limits by revisiting the concept and using derivatives to find some new limits.

Next, we will look at limits. Here is one. Let x equal expo to 0 of 3x over x.

The exponent on x is negative 1, so it is a reciprocal, or inverse. This leaves us with 3. Therefore, the limit is 3. All right, feeling smart? Now let's look at another one!

The limit of x to the 0th power over x is equal to x divided by x, when the exponent is 0. Because this simplifies to just x, the limit of x to the 0th power over x is equal to 0.

Since the limit of e to the x is 0, we can rewrite the expression as 2x over e to the x minus 1.

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Since the limit of this expression is also 0, there is no way to factor or reduce it further. Now, nothing reduces when x is equal to 0, so we must move on.

Let's look at the graph of each function separately.

In red, I have e to the x minus 1. In blue, I have 2x.

Thus, we can think of the ratio as not necessarily a rational expression but rather as the y-value of one function divided by the y-value of another.

If you zoom in on the x-axis at x equals 0, you will notice that as you zoom in further, the lines get closer and closer to being linear. The derivative!

So the blue line, our 2x, well, that was linear. So we know what the slope is. That's a slope of 2.

We will call the slope of our blue line f, and the slope of the red line g. The blue line has a slope of 2x, and the red line's slope is less than 2.

It appears to be at a 45-degree angle. The graph of e to the x appears to be a straight line for small values of x.

We found that for small values of x, e to the x equals 1 plus x. But then we subtracted the 1, so it more closely resembles x.

So the function's derivative behaves like two times the function's argument over x, which looks like the following problem. We know that the derivative is two, so there you go: limit equals two.

We can solve a derivative equation as follows: We say that the derivative of x is f(x) and the derivative of y is g(x). We then say that the limit of the y values equals the limit of the slope, which tells me that if I have-- this is important-- L'Hopital's Rule.

L'Hopital's Rule states that as x approaches some number for f of x over g of x, the limit as x approaches that number for the derivatives is equal to the ratio of the derivatives.

If the limit of a function is indeterminate. Be careful.

L'Hopital's Rule applies only to indeterminate forms of expressions involving limits. It does not apply to situations in which you plug in numbers and make a calculation.

If a function is approaching an asymptote, and you can't find the limit at first, take the derivative of the top, take the derivative at the bottom, and do the derivative again.