Find? #lim xrarr c ((x^3 - c^3)/ (x-c))#

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Answer 1 · 2018-06-17T10:32:12

#3c^2#

#lim_(x->c)((x^3-c^3)/(x-c))#

First, factor #x^3-c^3#
#x^3-c^3=(x-c)(x^2+xc+c^2)#

#lim_(x->c)(((x-c)(x^2+xc+c^2))/(x-c))#
#<=>lim_(x->c)(x^2+xc+c^2)#
#<=>c^2+c*c+c^2#
#<=>3c^2#

Answer 2 · 2018-06-18T05:14:02

# 3c^2#.

If one is familiar with the following Standard Form of Limit :

#lim_(x to a)(x^n-a^n)/(x-a)=na^(n-1)#,

then, the required limit #3c^2# follows immediately.

Here is another way to get the limit :

Let, #x=c+h," so that, (x-c)=h, and, as "x to c, h to 0#.

Further, #(x^3-c^3)/(x-c)=((c+h)^3-c^3)/h#,

#={cancel(c^3)+h^3+3ch(c+h)cancel(-c^3)}/h#,

#=[cancel(h){h^2+3c(c+h)}]/cancel(h)#,

#:."The Reqd. Lim."=lim_(h to 0) {h^2+3c(c+h)}#,

#=0^2+3c(c+0)#,

#=3c^2#, as Martin C. has readily derived!