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

Question

2476 views around the world

You can reuse this answer
Creative Commons License

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!

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