a problem about limits?

Question

It's the limit as x approaches -1 of [x^4 - x^(1/4)] / [x^3+x^(1/3)]

it is a limit problem from my high school textbook and it is about the theorem thats says "limit as x approaches a of [x^n - a^n] / [x^m - a^m] = (n/m)*[a^(n-m)] .

If someone could help me, please ?
and thanks

Answer 1

Over the real numbers, this limit does not exist, because the fourth root of -1 is not real-valued. The actual fourth roots of -1 are the four complex numbers

(sqrt (1/2)) X ( (plus or minus 1) + (plus or minus i)).

Over the complex numbers, since the denominator approaches
-2 as x tends to -1, and since the principal value of the fourth root of -1 is

(sqrt (1/2)) X (1 + i),

the limit would be -(sqrt (1/8)) X (1 + i).

Since this is a high school class, your teacher and text probably want the answer that the limit does not exist.

Answer 2

You can calculate the limit by plugging x = -1 directly into your formula. Then you get: (-1)^4 = 1, (-1)^(1/4) = e^(i*pi/4) = cos(pi/4) + i*sin(pi/4) = (1 + i )*sqrt(2)/2, and for denominator (-1)^3 = -1 and (-1)^(1/3) = -1 so denominator is -2.
Total expression (your limit) is then (1 - (1 + i)*sqrt(2)/2)/(-2).

Answer 3

If we plug x = a into [x^n - a^n] / [x^m - a^m], we get
[a^n - a^n] / [a^m - a^m] = 0 / 0 which is not acceptable
Factorizing both denominator and numerator,
= [(x - a){x^(n-1) + x^(n-2).a + ... ... + a^(n-1)}] / [(x - a){(x^(m-1) + x^(m-2).a + ... ... + a^(m-1)]}

Cancelling (x - a) from numerator and denominator
= {x^(n-1) + x^(n-2).a + ... ... + a^(n-1)}] / [(x - a){(x^(m-1) + x^(m-2).a + ... ... + a^(m-1)]}
Now plugging x = a,
= {a^(n-1) + a^(n-1) + a^(n-1) + ... ... + a^(n-1)} / {a^(m-1) + a^(m-1) + a^(m-1) + ... ... + a^(m-1)}
= {n times a^(n-1)} / {m times a^{m-1)
= (n/m) [a^(n-m)]

First part:
4th root of -1 is not possible. You have to deal with square root of -1.and power index is not ingeger.
OR please post your question again in detail

Answer 4

#2/ z=x/a then z - - > 1 and {a^n * (z^n – 1)} / {a^m * (z^m –1)} = l’hopital = [a^(n-m)]*[n*z^(n-1)] / [m*z^(m-1)] = (n/m) * a^(n-m)