What is the lim of h if it equals 0 Sinxcosh plus cosxsinh minus sinx divided by h?

Answer 1

lim(h→0) (sin x cos h + cos x sin h - sin x)/h

As h tends to 0, both the numerator and the denominator have limit zero. Thus, the quotient is indeterminate at 0 and of the form 0/0. Therefore, we apply l'Hopital's Rule and the limit equals:

lim(h→0) (sin x cos h + cos x sin h - sin x)/h

= lim(h→0) (sin x cos h + cos x sin h - sin x)'/h'

= lim(h→0) [[(cos x)(cos h) + (sin x)(-sin h)] + [(-sin x)(sin h) + (cos x)(cos h)] - cos x]]/0

= cosx/0 = ∞