What is this limit?

Question

lim [(1^x) / (1^x)]
x-->∞

1, indeterminate, or neither?

Everyone says 1, but that doesn't make much sense because the limit could also be written like this:
[lim(x-->∞) 1^x] / [lim(x-->∞) 1^x]

Right?

Answer 1

The rule of limit:
Simplify expression before whenever you need to evaluate the limit.

Thus lim [(1^x) / (1^x)] = lim [1] = 1.

§
Pertaining your other question...
Another property of the limit.
The lim f(x) = M and lim g(x) = N
iff lim f(x)/g(x) = M/N ... provided M and N are real numbers and N is not 0.

Thus you can only separate the limits if both parts can be expressed as a limit of real numbers. Otherwise the separation will not hold.

Answer 2

Hi Kyle,

It does make perfect sense. Agreed, you can write it as
[lim(x-->∞) 1^x] / [lim(x-->∞) 1^x]
but since lim(x-->∞) 1^x = lim(x->∞) 1 = 1 by itself, we get 1/1 again, so the answer is still 1.

Major Bob: 1^x = 1 for all real x, even x = 0. It is only 0^x that is undefined at x = 0. For any positive real number r, r^x is defined for all real x [r^x = e^(x ln r), which is defined when x ln r is, i.e. when r > 0 and for any x].

Answer 3

Your expression conveniently simplifies to 1/1, or 1, before you even need to think about its limit.

Answer 4

lim [(1^x) / (1^x)]=lim 1

x-->∞

Answer 5

Pretty sure it's 1.

Answer 6

1 to any power other than zero is 1, at zero it is undefined. in this case you will always have the limit of 1 regardless because the fraction will always equal 1.

Answer 7

it is one.