Simple calculus question

lol, I haven't taken calc in years.

What's the answer to this question?

lim(x -> 0) [1 / x]

Is it +-Inf? +Inf? -Inf?

Limit As x -> 0+ (from positive side) = +Inf

Limit As x-> 0- (from negative side) = -Inf

The simple answer is that it doesn't exist since the limits as x approaches zero from the left side does not equal the limit as x approaches zero from the right side. If you graph it, you can see this much more easily.
http://revisioncentre.co.uk/gcse/maths/1overx.gif

Edit: wait.... what the hell am I saying simple answer... it's the only answer 0.o

Strictly speaking, lim(x -> y) will never be equal for any limit as long the function's a "curve" (that is, f(x) = b, then lim(x -> y) = lim(x -> z). May be avoided by taking tops or bottoms of Sine or Cosine curves as well.)

But actually, the answer is Infinity. Cause ey, what's Infinity multiplied by -1?

Edit: Wait, is it undefined?

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Originally Posted by Devil's Sunrise

Strictly speaking, lim(x -> y) will never be equal for any limit as long the function's a "curve" (that is, f(x) = b, then lim(x -> y) = lim(x -> z). May be avoided by taking tops or bottoms of Sine or Cosine curves as well.)

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...What? This doesn't make any sense. For one, what is lim(x -> y) not supposed to equal? You never say.

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Originally Posted by Devil's Sunrise

But actually, the answer is Infinity. Cause ey, what's Infinity multiplied by -1?

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Negative infinity, which is a different number. :f6:

(Unless you're in a special geometry, and we aren't.)



Long story short, the answer is DNE, which is standard shorthand for Does Not Exist. The limit approaches two different values from either side of the point you're considering, so essentially, it doesn't "approach" anything at all in the sense you want... hence, Does Not Exist. Watch Mean Girls if you want Lindsey Lohan to confirm it for you... in an overly excited, valley girl kind of way. :wink:

EDIT: The answer isn't undefined. We say something is undefined when the question has been posed wrong... That's why 1/0 is undefined, because our system doesn't allow dividing by 0 anyway. This limit is a perfectly well-posed "question", it's just that the function in it happens to not be well-behaved at the point we're looking for.

That's why the term DNE exists. It's a fine point, though... The sort of thing math majors are interested in.

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Originally Posted by Morgana

...What? This doesn't make any sense. For one, what is lim(x -> y) not supposed to equal? You never say.

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Baww, bad example. I meant more like. Uh. Say we have a function f(x) = x. Then lim(x -> a) should be f(a + epsilon) = f(a - epsilon) where you let epsilon go towards 0. Then that will never happen unless f(x) = k or that f(x) = k * sin or k * cosin and that x is an element in the value handed to sin or cosin.

And bahh. Infinity isn't a number.

Quite simply speaking, you can define the answer in terms of approach, or L'Hospital's rule. For general purposes and derivatives/integrals, if all approaches do not equal each other, the particular point in question is undefined. For example, tan(90 degrees) is undefined.

@ Morgana, I've only seen DNE in L'Hospital shyts, and nowhere else o.O

I'm positive the answer is DNE...if that's not confirmed yet.

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Originally Posted by Devil's Sunrise

Baww, bad example. I meant more like. Uh. Say we have a function f(x) = x. Then lim(x -> a) should be f(a + epsilon) = f(a - epsilon) where you let epsilon go towards 0. Then that will never happen unless f(x) = k or that f(x) = k * sin or k * cosin and that x is an element in the value handed to sin or cosin.

And bahh. Infinity isn't a number.

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Fine, negative infinity is a different concept. Yeesh.

I don't understand your first point. You don't define what "k" is and I don't know how f(x) can equal a sine or cosine function when you already said f(x) = x. o.O And I don't get what you're trying to say with the trig anyway.

Does this help?
http://i35.photobucket.com/albums/d1...ey/limit-1.gif
In this case, the lim(x -> a) = 1, not 1.5. It doesn't matter what f(x) is at the point, it matters what it approaches.

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Originally Posted by KajitiSouls

Quite simply speaking, you can define the answer in terms of approach, or L'Hospital's rule. For general purposes and derivatives/integrals, if all approaches do not equal each other, the particular point in question is undefined. For example, tan(90 degrees) is undefined.

@ Morgana, I've only seen DNE in L'Hospital shyts, and nowhere else o.O

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I agree with you... the function tan(90 degrees) is undefined, since the function has no value at that point. But the limit of tan(x) as x approaches 90 degrees is DNE. Does that help?



Also, srsly guys, it's DNE. o.O

Quote:

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Originally Posted by KajitiSouls

Quite simply speaking, you can define the answer in terms of approach, or L'Hospital's rule. For general purposes and derivatives/integrals, if all approaches do not equal each other, the particular point in question is undefined. For example, tan(90 degrees) is undefined.

@ Morgana, I've only seen DNE in L'Hospital shyts, and nowhere else o.O

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Uh. What? I may be missing the point of your post, but L'Hopital's rule isn't applicable in this scenario. It's DNE because of what shouri posted.

Meh, I'm sorry. I meant f(x) = z, where z may be whatever you wish (ax + b, k and so forth)

k is a constant. Be it 1, 0, 90 or 1023. f(x) = x was just an example to see that f(a + epsilon) =/= f(a - epsilon), apart from when f(x) = k or when f(x) = k * (co)sin.

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Originally Posted by Zaotsu

Uh. What? I may be missing the point of your post, but L'Hopital's rule isn't applicable in this scenario. It's DNE because of what shouri posted.

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Well, fk you morning, look what you made me type up D=
(I'm not a morning person, I am a fking zombie until 11 am)

Um yeah, 1/x is not an indeterminate form xD But I still stand by my reasoning that the limit as x approaches 0 DNE (what Fiel is asking for), but in most other context, the function is undefined at x = 0.

The tangent thing you guys are talking about is similar to Fiel's question. tan(x) = sin(x)/cos(x), thus as x->90 degrees or (pi/2 rad), the function becomes undefined, thus the limit DNE.

I also have no flue what devil's getting at.

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Originally Posted by XTOTHEL

The tangent thing you guys are talking about is similar to Fiel's question. tan(x) = sin(x)/cos(x), thus as x->90 degrees or (pi/2 rad), the function becomes undefined, thus the limit DNE.

I also have no flue what devil's getting at.

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Okay then, ignore me, lol.