Question

$\lim_{ x-1 } x3 - 3n/x +1$

Answer 1

Answer:

limx→1[11−x−31−x3]=−1 and limx→1xn−1x−1=n

Explanation:

First Limit

limx→1[11−x−31−x3]
We cannot use the quotient rule for limits, because the denominator has a limit of 0 .

Write as a single ratio.
I will use (1−x3)=(1−x)(1+x+x2) (difference of cubes).

limx→1[11−x−31−x3]=limx→1[1+x+x2(1−x)(1+x+x2)−31−x3]

=limx→1(1+x+x2)−31−x3 =limx→1[−x2+x−2x3−1] =limx→1[−(x+2)(x−1)(x−1)(x2+x+1)] =limx→1[−x+2x2+x+1] =−33=−1

Second Limit

limx→1xn−1x−1
We cannot use the quotient rule for limits, because the denominator has a limit of 0 .

Because 1 is a zero of the numerator, x−1 is a factor of the numerator. Either recall the factorization or use division to get:
xn−1=(x−1)(xn−1+xn−2+⋅⋅⋅+x2+x+1)

So,

limx→1xn−1x−1=limx→1(xn−1+xn−2+⋅⋅⋅+x2+x+1)

There are n terms in the expression, each evaluating to 1 , so, we get

=1+1+1+⋅⋅⋅+1n terms =n