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/sci/ - Science & Math

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6154227 No.6154227 [Reply] [Original]

Why is the limit of ((e^x)-1)/x=1 as x approaches 0?
If I graph it, I can see that this is true, but why?

>> No.6154234

>>6154227
<span class="math">(e^{x})-1)[/spoiler] got a limit of 0 for x->0. Be 0 < x < 1.
Obviously, <span class="math"> (e^{x})-1)\div x > (e^{x})-1)[/spoiler].

I'm not gonna explain it further, but that should be a decent start.

>> No.6154238

>>6154234
Forgot some parenthesis. Still, it's readable.

>> No.6154259

You can use L'Hospital since it's a limit <span class="math">\frac{0}{0}[/spoiler]

<div class="math">\lim_{x \rightarrow 0} = \frac{\left(\frac{d}{dx}\left(e^x-1 \right)\right)}{\left(\frac{d}{dx}x\right)}=\lim_{x \rightarrow 0} \frac{e^x}{1} = 1</div>

>> No.6154263

>>6154259

You're making use of the derivative of e^x, which by definition is the limit of <div class="math">\frac{e^h-1}{h}</div> as h goes to zero, which is the result OP is trying to prove. Using L'Hopital is circular.

>> No.6154279

>>6154227

Use the limit definition of e:
<span class="math"> lim\limits_{x \rightarrow 0} (1+x)^{\frac{1}{x}} [/spoiler]

>> No.6154281

>>6154279
<span class="math"> \lim_{x \to 0} (1+x)^{\frac{1}{x}} [/spoiler]

>> No.6154283

>>6154259
This is correct and the quickest method, as both numerator and denominator are continuous.
>>6154263
This is retarded.

>> No.6154294

>>6154227
Do a Taylor Expansion.
e^x = 1 + x + x^2/2! + x^3/3! + ...
(e^x-1)/x = 1+x/2!+x^2/3!+...
Limit of this as x->0 is obviously 1, since all but the first term vanish.

>> No.6154730

>>6154283
>>6154263
>This is retarded.

It really depends of the context.
Using l'Hôpital is pretty stupid here, why not just notice that
((e^x)-1)/x is [exp(x)-exp(0)]/[x-0]
that goes to exp'(0) by the very definition of derivative (as >>6154263 said) ?
(If you're allowed to use the "result" that exp'=exp), you get your 1.


>>6154294
is also valid, but you need to know about taylor expansions...

All in all, what does OP know about exp ? What is his definition. There are several ones.

Without knowing the allowed hypo, you can't qualify a kind of "valid" answer as a retarded one...

>> No.6155077

>>6154283
get a load of this faggot