L'Hospital rule

This rule tells us that if we have a quotient [math]\displaystyle{ f(x)/g(x) }[/math] and the limits of each function converge to zero or to infinity for some point, then we can safely assume that the limit of the quotient [math]\displaystyle{ f/g }[/math] is equal to the limit of [math]\displaystyle{ f'/g' }[/math]. That is, the limit of the quotient of the derivatives. All under the condition that the functions are differentiable at that point. The graphical idea is that if both functions are differentiable, then near a point they both can be approximated by their respective linear approximations. In the same way that a linear approximation may make computations easier by replacing a complicated equation with a simpler one, L'Hospital rule replaces the algebraic manipulations that we'd have to do with calculating a quotient of derivatives. With this rule it becomes trivial to show that [math]\displaystyle{ \lim_{x \ \to \ 0} \frac{\sin(x)}{x} = 1 }[/math] for example.

Careful! This rule is exclusive to the specific cases of limits where we have [math]\displaystyle{ 0/0 }[/math] or [math]\displaystyle{ \infty/\infty }[/math]. Don't blindly use it thinking that we can use it anywhere. For example, consider [math]\displaystyle{ \lim_{x \ \to \ 0} \frac{x + 1}{x^2 + 2} }[/math]. We don't need L'Hospital and if we apply it, the result is going to be wrong.

One may naturally think that we can extend the rule to multivariable functions. We cannot do that because with multivariable functions we have partial derivatives, which represent rates of change on a per variable basis. We don't have an equivalent to the L'Hospítal rule for two or more variables. The reason for this lies in the fact that multiple paths lead to different limits in two or more variables. For multivariable functions there doesn't exist the concept of a derivative that is detached from direction. Suppose that we have a quotient [math]\displaystyle{ f(x,y)/g(x,y) }[/math] and an indeterminate form with a limit. We cannot consider [math]\displaystyle{ f'(x,y)/g'(x,y) }[/math] because there isn't the concept of differentiation in both variables at the same time.