L'Hopital's Rule: Using Derivatives to Evaluate Limits

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.
lim_(x->0) 3x/x=3
lim_(x->0)(x^2 -x)=0
lim_(x->0)
[Graph]
L'Hopital's Rule
lim_(x->a) f(x)/g(x)=lim_(x->a) f'(x)/g'(x)
if the limits an indeterminate form (0/0, inf/inf)
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 Oth 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 Oth 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 × minus 1. 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.

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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