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Calculus/Limits and functions

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Expert: - 2/7/2011


Question
Prove that lim[x:->a] [(x^n +a^n)/(x-a)]=na^(n-1)

Answer
Since the top and bottom of the fraction both go to 0,
the derivative of the top and bottom can be taken.
Even though a is the value x is approaching, it is still a constant and
therefore the derivative of it is 0.

The derivative of the top involves the derivative of x to a power, and this gives nx^(n-1).
The derivative of x-ais 1.  This gives [nx^(n-1)]/1 = nx^(n-1).

This is known as L'hospital's rule, as seen in http://mathworld.wolfram.com/LHospitalsRule.html

When a is put in for x, that is the limit.

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