lim s-0 (1+4s)^1/s

Question

lim s->0    (1+4s)^1/s

anonymous · Answer

I don't know what to do!. 1^0 is not an indeterminant form? Anybody please!??!

anonymous · Answer

=1/(1+4s)^s
=1/(1+4(0))^0
=1

anonymous · Answer

=1/(1+4s)^s
=1/(1+4(0))^0
=1

nikvist · Answer

$$\lim_{s\rightarrow 0}(1+4s)^{1/s}=\lim_{s\rightarrow 0}\left[(1+4s)^{1/4s}\right]^4=\left[\lim_{s\rightarrow0}(1+4s)^{1/4s}\right]^4=e^4$$

anonymous · Answer

Wow... nikvist is right

anonymous · Answer

The other two were wrong...

anonymous · Answer

WHy are they other 2 wrong?

jamesj · Answer

Nikvist is exactly right, because
$$ \lim_{x \rightarrow \infty} (1 + 1/x)^x = \lim_{x \rightarrow 0+} (1+x)^{1/x} = e $$

jamesj · Answer

the other two are wrong because they're applying a naive procedure that moves taking the limit inside functions (in this case exponential by the variable) which are simply false.  The fact that those results are not equal to the correct result demonstrate that the procedure they're using is mathematically invalid.

anonymous · Answer

Wouldnt x->0 (1+x)^1/x=1,  bc 1^infinity=1??

anonymous · Answer

You're awesome by the way, I can't thank you enough for the help!

jamesj · Answer

No, because implicit in what you've done is you've said
$$ \lim_{x \rightarrow 0} (1+x)^{1/x} = ( \lim_{x \rightarrow 0} (1+x) )^{\lim_{x \rightarrow 0} 1/x} $$
and that's just not valid.

anonymous · Answer

hmm, I don't understand what nikvist did in his first step. How did he get the exponent to 1/4s, and then why is their another exponent 4.

jamesj · Answer

Call the initial expression f(x).  Then he just wrote that

$$ ( \ (f(x))^{1/4} \ )^4 $$

anonymous · Answer

So he just added an exponent, just because? That still doesn't explain were he got the 1/4s instead of just 1/4 as an exponent

jamesj · Answer

$$ (1+4s)^{1/s} =   ( (1+4s)^{1/s} )^{(1/4).4} = ([ (1+4s)^{1/s} ]^{(1/4)} )^4 = ((1+4s)^{1/4s})^4 $$

anonymous · Answer

Rule of exponents question, when simplifying x^2^3, do you do 3*2 or 3+2?

anonymous · Answer

I think it's multiply, that would explain the 1/s * 1/4=1/4s.

Okay I get it now. So the trick is to realize e=(1+x)^1/x

Man that was tricky. I don't know if i would have realized that I needed to pull out the 4 from 4/4.