Question

What is the limit of (x^n-a^n)/(x-a)?

Answer 1

QUESTION⤵

What is the limit of (x^n-a^n)/(x-a)?

ANSWER ⤵

The answer will be na^n-1.

As (1+X)^n=1^nCo+nC1X+nC2X^2+.......nCnX^n

=lim. x^n -a^n/x-a

X➡a+

=lim. (a+h)^n - a^n/ (a+h) -a

h➡0

lim. [a(1+h/a]-a^n/h

h➡0

lim. [a^n(1+h/a] - a^n/h

h➡0

lim. a^n[(1+h/a)^n -1]/h

h➡0

lim. a^n[(1+h/a)^n -1]

h➡0

lim. a^n/h[1+ n of h/a+ n(n-1)/2! (h/a)^2+.......-1]

h➡0

lim. a^n/h[n of h/a +n(n-1)/2!(h/a)^2+......+nCn (h/a)^n]

h➡0

here, we will take h as common.

lim. a^n/h. h [n/a+n(n-1)/2!.h/a^2+....+h^n-1/a^n]

h➡0

we'll get,

a^n.n/a

n.a^n-1

${{\huge{\boxed{\mathcal{hope\:it\:helps}}}}}$