how does one evaluate the limit of:
Lim(x->0) (x-tan(x))/(x^3)

- inkyvoyd

how does one evaluate the limit of:
Lim(x->0) (x-tan(x))/(x^3)

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- schrodinger

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- inkyvoyd

\(\Huge \lim_{x \rightarrow 0}\frac{x-\tan x}{x^3}\)

- inkyvoyd

Wolfram says the answer is -1/3, but I'm more interested in a concise process - wolfram's evaluation steps are incredibly long and convoluted.

- hartnn

can u use L'Hopital's ?

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- anonymous

You sure can!
0-0 / 0 = 0
Wanna work on this together, inky?

- inkyvoyd

You can - but doing it the fast way does not involve l'hopital's - I tried it, and it's really not very efficient...

- anonymous

Oh! Well, let's do it manually, then!
Ready?

- inkyvoyd

First use of l'hopital's: \(\Huge \lim_{x \rightarrow 0}\frac{1-sec^2(x)}{3x^2}\)

- inkyvoyd

Ready.

- inkyvoyd

use of trig identities:
\(\Huge \lim_{x \rightarrow 0} \frac{\tan^2(x)}{3x^2}\)

- inkyvoyd

Erm, that should be a negative.

- inkyvoyd

\(\Huge -\lim_{x \rightarrow 0} \frac{\tan^2(x)}{3x^2}\)

- anonymous

I am a tad bit embarrassed that I am admitting this, but I want to make sure to be on the safe side (it's been a while since I differentiated):
You can legally add two halves of a differentiation to get the final answer, right?
As in split it between differentiation of (x/x^3) - (tanx)/x^3 ?
I ask this because it's been about a year since cal 1, and I'm in cal 2 right now and am pretty much only doing integrations, lol.

- inkyvoyd

Yeah, you can split up the limit - the only problem is that we get two indeterminant forms instead of one, resulting in more chaos lol

- inkyvoyd

I'm wondering if I should go ahead and just use l'hopital's twice more...

- anonymous

Yeah, we can definitely do that. I thought we were going to skip that lolz. Looking good so far.

- inkyvoyd

The only problem is that I'm pretty sure I'm that follows wolfram's steps.

- anonymous

(tanx)^2 / (3x^2)
Oh, it's ok, I think we'll be able to solve this eventually. Worst case scenario, we can fall back to that other method of manually doing the derivatives.
Let's see where this gets us if we keep going.

- inkyvoyd

Do you still remember the sandwich theorem?

- anonymous

Oh dear, not at all. I was horrible at that one. Is this one of those instances where we need it? :/

- inkyvoyd

I'm pretty sure it would help, but I haven't gotten to it.
My course (AP calc BC) just started on the "intuitive definition of limits", and want us to use a graphing calculator to evaluate limits by eyeballing it. My thoughts on that are
- screw that, I didn't skim over a calc book last year to be spoonfed into doing limits the wrong way.
So yeah, I pushed myself into an awkward situation with this limit.

- anonymous

-lim x->0 ( 2(tanx)*sec^2(x) ) / (6x)
I like that your style, though. I admit I was lazy in cal 1, so that's why I'm so weak. But ironically, I'm a beast at Cal 2. I can integrate almost any reasonable expression and can find areas under a curve like it was a simple long division question. ;)

- anonymous

Woo! I think this next one will solve it for us!

- anonymous

Well, assuming sec(0) gives us something besides undefined...

- anonymous

Heck ya! We're good to go (it's 1).

- inkyvoyd

I think on this step wolf uses l'hopital's again but gets some factor of x on the bottom and screws everything up

- anonymous

-lim x->0 ( 2(tanx)*sec^2(x) ) / (6x)
-2/6 lim x -> 0 [tanxsec^2x]/[x]
Just one more l'hopital left!

- anonymous

We should be ok, though, because that x will turn into a 1. It's a matter of correctly chainruling the top, so correct me if I goof up (I will :P)

- inkyvoyd

Lol this is probably going to be the worst derivative I've had in days - at least we have always have rules for those ;)

- anonymous

denominator: (tanx)(secx)^2
(secx^2 * 2 (secx) * tanxsecx)/1
2 * (secx)^2 * tanx * secx
2 * (secx)^3 * tanx
So...
-2/6 lim x -> 0 2 * (secx)^3 * tanx
-4/6 lim x -> 0 tanx(secx)^3
-2/3 lim x -> 0 tanx(sec^x)^3

- anonymous

Does this look wrong?

- inkyvoyd

I got
tan(x)(2 sec^2(x) tan(x))+sec^4(x) for the numerator?

- anonymous

Hmm.... let's work this out individually on paper and see if we get the same answer. I'm curious on this one and wanna get an answer. :P
So wanna start from scratch and do it on paper and compare after every l'hopital?

- inkyvoyd

Sure.

- inkyvoyd

Shall we post step by step or all steps?

- anonymous

I guess at the end of each l'hopital and then discuss if we differ.
I guess I got too excited and went almost all the way. :P

- inkyvoyd

same lol

- anonymous

After my first lhopital:
-lim (1 - sec^2 x) / (3x^2)

- anonymous

dang it... I mixed two steps together. Ignore the negative limit.

- inkyvoyd

First step
\(\Huge \lim_{x \rightarrow 0}\frac{1-\sec^2 x}{3x^2}\)

- inkyvoyd

Mm, then both are the same.

- anonymous

Then I simplified into:
-1/3 lim (tan^2 x)/(x^2)
This should be legitimate. (No new lhopital yet)

- inkyvoyd

Second step:
\(\Huge -\frac{1}{3}\lim_{x \rightarrow 0} \frac{\tan^2(x)}{x^2}\)

- inkyvoyd

Yerp.

- anonymous

Basically, just pulled out the 3 from the denominator if it's not too clear.

- anonymous

Oh nice. Thinkin' alike. I like.

- anonymous

So then after next lhopital:
-1/3 lim [(tanx)(secx)^2]/x
(I canceled a 2 in both the numerator and denominator)

- inkyvoyd

That's what I got - This whole TeX thing is messing with my typing speed lol

- inkyvoyd

Well I gotta finish this next step...

- anonymous

No prob; I can't complain, considering how ugly my writing is. :P

- anonymous

Next up (let's do this step by step to be safe):

- anonymous

the tanx turns into (secx)^2

- anonymous

Fair enough so far?

- inkyvoyd

I'm unreasonably bad at the chain rule, so I split it up into two product rules

- anonymous

(Keep a careful eye on my work; I might accidentally integrate instead :P)

- anonymous

Ah, ok. We should eventually arrive at the same answer, then. I'll keep typing out my steps in hopes that you might catch an obvious error by me.

- inkyvoyd

Alrighty.

- anonymous

So for my (secx)^2, I dropped the 2 down in front of the secx:
2(sexc)^1
And then chainruled the inside:
2(secx)(secxtanx)
I think we should have both gotten (secx)^2 * 2 * secx * secx * tanx for our numerators. Got anything similar? (Haven't simplified yet)

- inkyvoyd

yup that's what product rule yields.

- anonymous

The denominator turned into a 1 for me (it was only an x before lhopitaling)

- inkyvoyd

btw, I'll start typing up my result with product rule

- anonymous

Okie dokie; I'll start cleaning up the numerator.

- inkyvoyd

But only the numerator, no more limit crap and fractions lol, that stuff wastes so much time

- anonymous

(secx)^2 * 2 * secx * secx * tanx
= secx * secx * 2 * secx * secx * tanx
= 2 * (secx)^4 * tanx

- anonymous

Aww man... but there's a two just begging to into the limit section. ;)

- anonymous

Oh nice. In one hour I'll have been up officially for 24 straight hours; roughly 15 of them spent on studying/math fun. :P

- inkyvoyd

start out
\(\Huge \tan(x)\sec^2(x)\)
\(\Huge \tan(x)\frac{d\sec^2(x)}{dx}+\sec^4x\)
\(\Huge \tan(x)(2\tan(x)\sec^2(x))+\sec^4x\)
\(\Huge \tan(x)(2\tan(x)\sec^2(x))+\sec^4x\)
Then we put in 0 for x, cause the denom is already reduced to one
\(\Huge 0(\text{blahblahblahblahblah})+1^4\)
\(\Huge 1\)
Multiply by what's out of the limit
\(\Huge -\frac{1}{3}\)
Rawrr.

- anonymous

Whoa. The site was dying/freezing on me there. Brings back memories of how this site used to be a year ago. lolz.

- inkyvoyd

Lol - you know me don't you o.o

- anonymous

In real life? I doubt it. But we may have been friends on this site on the past (no offense or anything, but I don't recall your nickname).

- inkyvoyd

omg the next problem I'm supposed to evaluate with the calculator is freakin insane.

- inkyvoyd

\(\Huge \lim_{x \rightarrow 0}(1+x)^{\frac{1}{x}}\)

- inkyvoyd

This one's either e or 1/e

- anonymous

One question, though: whenever I did mine, it came out to like 0, I think. Where'd your +sec^4 come from? I wanna learn that.

- anonymous

OH! Product rule, duh! Sorry.

- inkyvoyd

lol I really have to stop using Tex, was 90% done then you figured it out xD

- anonymous

Lolz, sorry. I was just curious since if I put in my x values on my numerator, I'd get 0. :P But all good now. Alright, let's do this! LEEEERRRROOOYYYYYYYy JENNNKINSSS!

- anonymous

Alright, let's see if I've still got this in me:

- inkyvoyd

Lol, I'm outta time.
But at least I have chicken.
(not really)

- anonymous

Start by doing
e^ln(that crap)

- inkyvoyd

Well I got 3 more mins or so

- anonymous

^ Does the reasoning behind that make sense?

- anonymous

Aw man. Didn't realize you were timed. :(

- inkyvoyd

I think 3 mins is enough though :o

- inkyvoyd

and yes, it does make sense

- anonymous

alright. Got it to here:
e to ln of (1+x)/x
Try to solve that on paper really fast if you can. Get it to 0/0 or inf/inf to use lhopital.

- inkyvoyd

btw why is it justifiable that we can put the limit onto the exponent?

- anonymous

The limit is technically a variable so you can screw it it. You can multiply it, divide it, treat it as an exponent, etc.

- inkyvoyd

got it

- anonymous

So let's see:
[ln(1+x)]/x
e^
This is what I should have told you to write. And yes, the x is dividing the ln(1+x). This should be lhopitaable.

- inkyvoyd

ln(1+x)/x
(1/(x+1)*1)/1
1

- anonymous

me too.

- inkyvoyd

Well bell rang, so I'll clean up and maybe see you later. Thanks for the help!

- anonymous

So e^1, I think. (Don't forget the e :P)
Sounds good; have fun. I'm going to take a 4 hour lap before going to class I guess. :P
I'll fan ya, maybe you can toss me some more probs; these were fun. See ya.

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