what is the limit as h approaches 0 of
((x+h)^pi-x^pi)/h?
Multiple choice answers:
A: lnx
B: lnx (x^pi)
C: Pi lnx
D: (pi^2) ln (x^2)
E: (pi^x) lnx
I know this is d/dx x^pi, or Pi x^(Pi-1)
You are correct. I fear that the answer choices relate to a different problem, since with a simple power, no logs will arise.
To find the limit as h approaches 0 of the given expression, we can start by simplifying the expression:
((x+h)^pi - x^pi) / h
To simplify, we can expand the numerator using the binomial theorem:
((x+h)^pi - x^pi) = x^pi + pi * x^(pi-1) * h + higher order terms - x^pi
Notice that the x^pi terms cancel out, leaving us with:
pi * x^(pi-1) * h / h
The h in the numerator and the denominator cancel out, leaving us with:
pi * x^(pi-1)
Therefore, the limit as h approaches 0 is pi * x^(pi-1).
Now, let's examine the multiple-choice answers:
A: lnx
B: lnx (x^pi)
C: Pi lnx
D: (pi^2) ln (x^2)
E: (pi^x) lnx
Comparing our simplification with the given choices, we see that none of the choices match our derived result of pi * x^(pi-1). Therefore, none of the provided choices are correct.