Show #lim_ (x->2) {(1+x)^n-3^n}/(x-2)=n.3^(n-1)#?

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Answer 1 · 2017-12-05T09:06:38

See below.

Making #x + 1= 3+h rArr x = 2+h# and substituting

#lim_(x->2){(1+x)^n-3^n}/(x-2) = lim_(h->0)((3+h)^n-3^n)/h#

but

#lim_(h->0)(f(x+h)-f(x))/h = f'(x)# then

#lim_(h->0)((3+h)^n-3^n)/h = n 3^(n-1)#

NOTE

#((3+h)^n-3^n)/h = (n 3^(n-1)h + (n(n-1))/(2!)3^(n-2)h^2+cdots + h^n)/h = n 3^(n-1) + (n(n-1))/(2!)3^(n-2)h+cdots + h^(n-1) = n3^(n-1)+h(p(h))#