L'Hospital rule: Difference between revisions
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One may naturally think that we can extend the rule to multivariable. 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. Theoretically, we'd need a very special condition which would be both limits, for the numerator and denominator, to converge to infinity or zero for all possible paths at the same time. With what we know for multivariable functions, the indeterminate forms <math>0/0</math> and <math>\infty/\infty</math> depend on which path we take. | 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. Theoretically, we'd need a very special condition which would be both limits, for the numerator and denominator, to converge to infinity or zero for all possible paths at the same time. With what we know for multivariable functions, the indeterminate forms <math>0/0</math> and <math>\infty/\infty</math> depend on which path we take. | ||
Revision as of 04:23, 28 March 2022
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. Theoretically, we'd need a very special condition which would be both limits, for the numerator and denominator, to converge to infinity or zero for all possible paths at the same time. With what we know for multivariable functions, the indeterminate forms [math]\displaystyle{ 0/0 }[/math] and [math]\displaystyle{ \infty/\infty }[/math] depend on which path we take.
