- #1

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$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$

From this, how can I get the definition of

$$\lim_{x\to p^-}=-\infty? $$

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- Thread starter RubroCP
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In summary, the conversation discusses the definition of a limit where the function approaches positive or negative infinity. The definition states that for the limit to be positive infinity, the function values can be made arbitrarily large around the limit point, and for the limit to be negative infinity, the function values can be made arbitrarily small around the limit point. The conversation also briefly touches on the concept of cardinality and its relevance to limit values.f

- #1

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$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$

From this, how can I get the definition of

$$\lim_{x\to p^-}=-\infty? $$

- #2

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Now, how does this transform to low instead of high?

- #3

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Thanks, my friend. I will try here again.

Now, how does this transform to low instead of high?

- #4

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The usual definitions don't use ##\epsilon## for this type of limit, but instead use M or some other letter. ##\epsilon## is typically some small number, relatively close to zero.

$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$

From this, how can I get the definition of

$$\lim_{x\to p^-}=-\infty? $$

The definition you have usually goes like this:

$$ \lim_{x\to p^+}f(x)=+\infty\iff \forall M >> 0, \exists \delta > 0, \text{ if } p \lt x \lt p + \delta \text{ then } f(x) > M$$

The definition for the limit being negative infinity is similar.

- #5

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What about the "limit" of 1/x as x tends to 0? There is no interval around 0 where 1/x is >100.The definition you have usually goes like this:

limx→p+f(x)=+∞⟺∀M>>0,∃δ>0, if p<x<p+δ then f(x)>M

The definition for the limit being negative infinity is similar.

- #6

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The definition I wrote is for a one-sided limit. For any x in the half-interval (0, .01), 1/x > 100.What about the "limit" of 1/x as x tends to 0? There is no interval around 0 where 1/x is >100.

- #7

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Since my main background is complex analysis, I prefer the Riemann sphere mapping.

- #8

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Saying that ##\lim_{x \to x_0} f(x) = \infty## is merely shorthand for saying that the function values get arbitrarily large as x gets nearer to ##x_0##. The function that you listed, f(x) = 1/x, can be made arbitrarily large and positive for positive x near zero, and can be made arbitrarily negative for negative x near zero. Neither limit "exists" in the sense of being a real number, but the infinity symbol conveys the unboundedness of the function values. The two one-sided limits are qualitatively and quantitatively about as different as they could possibly be, so, yes, ##+\infty## and ##-\infty## are very different.OK. I would prefer using absolute values, though. And then there is the philosophical question: Is +∞ different from -∞? Since neither one represents a number, how can we tell?

- #9

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https://en.wikipedia.org/wiki/CardinalityThe two one-sided limits are qualitatively and quantitatively about as different as they could possibly be, so, yes, +∞ and −∞ are very different.

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