Problem. Calculate
limit(cosh(sqrt(x+1))-cosh(sqrt(x)))^(1/sqrt(x)),x,inf).
see:
http://www.math.utexas.edu/pipermail/maxima/2012/030330.html
Direct calculation gives Maxima bug. We solve step-by-step
By mean value theorem
f(x+1)-f(x)=f'(c), x < c < x+1
If f'(x) is increasing, then f'(x)<f(x+1)-f(x)<f'(x+1) and
f'(x)^r<(f(x+1)-f(x))^r<f'(x+1) ^r
(%i1) f(x):=cosh(sqrt(x));
(%o1) f(x):=cosh(sqrt(x))
(%i2) define(f1(x),diff(f(x),x));
(%o2) f1(x):=sinh(sqrt(x))/(2*sqrt(x))
derivative f'(x)=f1(x) is increasing:
(%i3) wxplot2d([f1(x)], [x,10,100])$
(%t3) << Graphics >>
(%i4) r:1/sqrt(x)$
Then it is true inequality
(%i5) (f1(x)^r<(f(x+1)-f(x))^r)<f1(x+1)^r;
(%o5) sinh(sqrt(x))^(1/sqrt(x))/(x^(1/(2*sqrt(x)))*2^(1/sqrt(x))) <
(cosh(sqrt(x+1))-cosh(sqrt(x)))^(1/sqrt(x)) <
sinh(sqrt(x+1))^(1/sqrt(x))/((x+1)^(1/(2*sqrt(x)))*2^(1/sqrt(x)))
Limit left side is %e
(%i6) limit(f1(x)^r,x,inf);
(%o6) %e
Limit right side also is %e
(%i7) limit(f1(x+1)^r,x,inf);
(%o7)
limit(sinh(sqrt(x+1))^(1/sqrt(x))/((x+1)^(1/(2*sqrt(x)))*2^(1/sqrt(x))),x,inf)
(%i8) exponentialize(%),radcan$
(%i9) ev(%, nouns);
(%o9) %e
Then limit(cosh(sqrt(x+1))-cosh(sqrt(x)))^(1/sqrt(x)),x,inf)=%e
(%i10) build_info();
(%o10) build_info("5.28.0-2","2012-08-27 23:16:48","i686-pc-mingw32","GNU
Common Lisp (GCL)","GCL 2.6.8")
Best
Aleksas