Quandaries and Queries


Find the limit of [(1/(x+4))-(1/4)]/x as x approaches zero.
How do you use l"hopital's rule to find this limit. I know how to do it with multiplying everything by 4(x+4), and getting the answer, -1/16.But how do you apply derivatives with l'hopitals rule to this type of problem?


Hi Abraham,

I would do this problem the way you did, but if you insist on using l'hopital's rule you can do so. As x approaches zero both the numerator

 1/(x + 4)1/4

and the denominator


approach zero, so it is valid to apply l'hopital's rule.

The derivative of the numerator (using the quotient rule) is

0 - 1(1)/(x + 4)2-1/(x + 4)2

and the derivative of the deminator is 1, so one application of l'hopital's give the limit, as x approaches zero, of

 -1/(x + 4)2

This limit is clearly -1/16, so the value of the original limit is -1/16 also.



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