Limit - Revision history

Vipul: /* Right-hand limit */

2021-09-25T03:14:51Z

Right-hand limit

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	\| we have <math>\|f(x) - L\| < \varepsilon</math> \|\| <math>f(x) \in (L - \varepsilon,L + \varepsilon)</math> \|\| The symbol <math>\| \ \ \|</math> stands for the [[absolute value function]].<br><math>\in</math> stands for "is in the set" so the statement should be read as saying that <math>f(x)</math> is in the set <math>(L - \varepsilon, L + \varepsilon)</math>.		\| we have <math>\|f(x) - L\| < \varepsilon</math> \|\| <math>f(x) \in (L - \varepsilon,L + \varepsilon)</math> \|\| The symbol <math>\| \ \ \|</math> stands for the [[absolute value function]].<br><math>\in</math> stands for "is in the set" so the statement should be read as saying that <math>f(x)</math> is in the set <math>(L - \varepsilon, L + \varepsilon)</math>.
	\|}		\|}
	The '''right hand limit''' (acronym '''RHL''') <math>\lim_{x \to c^+} f(x)</math> is defined as a value <math>L \in \R</math> such that <math>\lim_{x \to c^+} f(x) = L</math>. By the [[uniqueness theorem for limits]] (one-sided version), there is at most one value of <math>L \in \R</math> for which <math>\lim_{x \to c^+} f(x) = L</matH>. Hence, it makes sense to talk of ''the'' right hand limit when it exists.
			The '''right-hand limit''' (acronym '''RHL''') <math>\lim_{x \to c^+} f(x)</math> is defined as a value <math>L \in \R</math> such that <math>\lim_{x \to c^+} f(x) = L</math>. By the [[uniqueness theorem for limits]] (one-sided version), there is at most one value of <math>L \in \R</math> for which <math>\lim_{x \to c^+} f(x) = L</matH>. Hence, it makes sense to talk of ''the'' right hand limit when it exists.

	===Side-by-side comparison of the definitions===		===Side-by-side comparison of the definitions===

Vipul: /* Left-hand limit */

2021-09-25T03:14:39Z

Left-hand limit

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	\|}		\|}

	The '''left hand limit''' (acronym '''LHL''') <math>\lim_{x \to c^-} f(x)</math> is defined as a value <math>L \in \R</math> such that <math>\lim_{x \to c^-} f(x) = L</math>. By the [[uniqueness theorem for limits]] (one-sided version), there is at most one value of <math>L \in \R</math> for which <math>\lim_{x \to c^-} f(x) = L</math>. Hence, it makes sense to talk of ''the'' left hand limit when it exists.		The '''left-hand limit''' (acronym '''LHL''') <math>\lim_{x \to c^-} f(x)</math> is defined as a value <math>L \in \R</math> such that <math>\lim_{x \to c^-} f(x) = L</math>. By the [[uniqueness theorem for limits]] (one-sided version), there is at most one value of <math>L \in \R</math> for which <math>\lim_{x \to c^-} f(x) = L</math>. Hence, it makes sense to talk of ''the'' left hand limit when it exists.

	===Right-hand limit===		===Right-hand limit===

Vipul: /* Right hand limit */

2021-09-25T03:14:07Z

Right hand limit

← Older revision		Revision as of 03:14, 25 September 2021
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	The '''left hand limit''' (acronym '''LHL''') <math>\lim_{x \to c^-} f(x)</math> is defined as a value <math>L \in \R</math> such that <math>\lim_{x \to c^-} f(x) = L</math>. By the [[uniqueness theorem for limits]] (one-sided version), there is at most one value of <math>L \in \R</math> for which <math>\lim_{x \to c^-} f(x) = L</math>. Hence, it makes sense to talk of ''the'' left hand limit when it exists.		The '''left hand limit''' (acronym '''LHL''') <math>\lim_{x \to c^-} f(x)</math> is defined as a value <math>L \in \R</math> such that <math>\lim_{x \to c^-} f(x) = L</math>. By the [[uniqueness theorem for limits]] (one-sided version), there is at most one value of <math>L \in \R</math> for which <math>\lim_{x \to c^-} f(x) = L</math>. Hence, it makes sense to talk of ''the'' left hand limit when it exists.

	===Right hand limit===		===Right-hand limit===

	Suppose <math>f</math> is a [[function]] of one variable and <math>c \in \R</math> is a point such that <math>f</math> is defined on the immediate right of <math>c</math> (note that <math>f</math> may or may not be defined at <math>c</math>). In other words, there exists some value <math>t > 0</math> such that <math>f</math> is defined on <math>(c,c+t)</math>.		Suppose <math>f</math> is a [[function]] of one variable and <math>c \in \R</math> is a point such that <math>f</math> is defined on the immediate right of <math>c</math> (note that <math>f</math> may or may not be defined at <math>c</math>). In other words, there exists some value <math>t > 0</math> such that <math>f</math> is defined on <math>(c,c+t)</math>.

Vipul: /* Left hand limit */

2021-09-25T03:13:57Z

Left hand limit

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	* In order to be able to talk of ''the'' limit <math>\lim_{x \to c} f(x)</math>, what additional fact do we need beyond the definition of what <math>\lim_{x \to c} f(x) = L</math> means?		* In order to be able to talk of ''the'' limit <math>\lim_{x \to c} f(x)</math>, what additional fact do we need beyond the definition of what <math>\lim_{x \to c} f(x) = L</math> means?

	===Left hand limit===		===Left-hand limit===

	Suppose <math>f</math> is a [[function]] of one variable and <math>c \in \R</math> is a point such that <math>f</math> is defined on the immediate left of <math>c</math> (note that <math>f</math> may or may not be defined at <math>c</math>). In other words, there exists some value <math>t > 0</math> such that <math>f</math> is defined on <math>(c-t,c)</math>.		Suppose <math>f</math> is a [[function]] of one variable and <math>c \in \R</math> is a point such that <math>f</math> is defined on the immediate left of <math>c</math> (note that <math>f</math> may or may not be defined at <math>c</math>). In other words, there exists some value <math>t > 0</math> such that <math>f</math> is defined on <math>(c-t,c)</math>.

Vipul: /* Prover's strategy revisited */

2015-05-16T16:14:46Z

Prover's strategy revisited

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	===Prover's strategy revisited===		===Prover's strategy revisited===

	The prover, in choosing a winning strategy, must specify a rule that can determine a value of <math>\delta</math> that works in terms of the value of <math>\varepsilon</math> specified by the skeptic. In other words, the prover must have a way of specifying <math>\delta</math> '''as a function of'' <math>\varepsilon</math>.		The prover, in choosing a winning strategy, must specify a rule that can determine a value of <math>\delta</math> that works in terms of the value of <math>\varepsilon</math> specified by the skeptic. In other words, the prover must have a way of specifying <math>\delta</math> ''as a function of'' <math>\varepsilon</math>.

	The skeptic also chooses <math>x</math> in the next move. However, the prover has no way of knowing the value of <math>x</math> that the skeptic plans to pick. Thus, in order for the prover to have a winning strategy, the prover's choice of <math>\delta</math> should be such that ''no matter what'' <math>x</math> the skeptic picks, the prover wins.		The skeptic also chooses <math>x</math> in the next move. However, the prover has no way of knowing the value of <math>x</math> that the skeptic plans to pick. Thus, in order for the prover to have a winning strategy, the prover's choice of <math>\delta</math> should be such that ''no matter what'' <math>x</math> the skeptic picks, the prover wins.

Vipul: /* Relation between the limit notions */

2015-05-16T16:12:07Z

Relation between the limit notions

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	The two-sided limit exists if and only if (both the left hand limit and right hand limit exist) '''and''' (they are equal to each other).		The two-sided limit exists if and only if (both the left hand limit and right hand limit exist) '''and''' (they are equal to each other).

	Explicitly, <math>\lim_{x \to c} f(x)</math> exists if all three of these conditions hold:		Explicitly, <math>\lim_{x \to c} f(x)</math> exists if '''all three''' of these conditions hold:

	* <math>\lim_{x \to c^-} f(x)</math> exists.		* <math>\lim_{x \to c^-} f(x)</math> exists.
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	* <math>\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)</math>.		* <math>\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)</math>.

	Moreover, in the event that both one-sided limits exist, the two-sided limit is equal to both of them.		Moreover, in the event that both one-sided limits exist and are equal, the two-sided limit is equal to both of them.

	Further, a particular value of <math>\delta > 0</math> works for a particular value of <math>\varepsilon > 0</math> in the two-sided limit definition if and only if it works in both the left hand limit definition and the right hand limit definition.		Further, a particular value of <math>\delta > 0</math> works for a particular value of <math>\varepsilon > 0</math> in the two-sided limit definition if and only if it works in both the left hand limit definition and the right hand limit definition.

Vipul at 00:32, 29 April 2014

2014-04-29T00:32:29Z

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	{{perspectives}}		{{perspectives}}
			{{core term}}
	==Motivation==		==Motivation==

Vipul: /* Definition of finite limit for function of one variable in terms of a game */

2013-09-27T18:52:07Z

Definition of finite limit for function of one variable in terms of a game

← Older revision		Revision as of 18:52, 27 September 2013
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	The game is between two players, a '''Prover''' whose goal is to prove that the limit statement is true, and a '''Skeptic''' (also called a '''Verifier''' or sometimes a '''Disprover''') whose goal is to show that the statement is false. The game has three moves:		The game is between two players, a '''Prover''' whose goal is to prove that the limit statement is true, and a '''Skeptic''' (also called a '''Verifier''' or sometimes a '''Disprover''') whose goal is to show that the statement is false. The game has three moves:

	# First, the skeptic chooses <math>\varepsilon > 0</math>, or equivalently, chooses the target interval <math>(L - \varepsilon,L + \varepsilon)</math>.		# First, the skeptic chooses <math>\varepsilon > 0</math>, or equivalently, chooses the target interval <math>(L - \varepsilon,L + \varepsilon)</math> in which the skeptic is challenging the prover to ''trap'' the function.
	# Then, the prover chooses <math>\delta > 0</math>, or equivalently, chooses the interval <math>(c - \delta, c + \delta) \setminus \{ c \}</math>.		# Then, the prover chooses <math>\delta > 0</math>, or equivalently, chooses the interval <math>(c - \delta, c + \delta) \setminus \{ c \}</math>.
	# Then, the skeptic chooses a value <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math>, or equivalently, <math>x \in (c - \delta, c + \delta) \setminus \{ c \}</math>, which is the same as <math>(c - \delta,c) \cup (c,c + \delta)</math>.		# Then, the skeptic chooses a value <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math>, or equivalently, <math>x \in (c - \delta, c + \delta) \setminus \{ c \}</math>, which is the same as <math>(c - \delta,c) \cup (c,c + \delta)</math>.
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	0:09:37.920,0:09:40.500		0:09:37.920,0:09:40.500
	I explain the mathematical symbols.</toggledisplay>		I explain the mathematical symbols.</toggledisplay>

			===Negation of limit statement and non-existence of limit===

	We now consider the explicit description of the definition for the case that the skeptic has a winning strategy for the limit game for <math>\lim_{x \to c} f(x) = L</math>, i.e., for the limit statement being false.		We now consider the explicit description of the definition for the case that the skeptic has a winning strategy for the limit game for <math>\lim_{x \to c} f(x) = L</math>, i.e., for the limit statement being false.

Vipul: /* Two-sided limit */

2013-09-08T17:12:12Z

Two-sided limit

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	We now consider the explicit description of the definition for the case that the skeptic has a winning strategy for the limit game for <math>\lim_{x \to c} f(x) = L</math>, i.e., for the limit statement being false.		We now consider the explicit description of the definition for the case that the skeptic has a winning strategy for the limit game for <math>\lim_{x \to c} f(x) = L</math>, i.e., for the limit statement being false.

			In words, the definition is:

			{{quotation\|There exists <math>\varepsilon > 0</math> such that for every <math>\delta > 0</math>, there exists <math>x</math> satisfying <math>0 < \|x - c\| < \delta</math> and <math>\|f(x) - L\| \ge \varepsilon</math>.}}

	Let's review the definition in conjunction with the game along with a deeper semantic understanding of the steps:		Let's review the definition in conjunction with the game along with a deeper semantic understanding of the steps:

Vipul: /* Two-sided limit */

2013-09-08T17:09:58Z

Two-sided limit

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	<math>\!\lim_{x \to c} f(x) = L</math>		<math>\!\lim_{x \to c} f(x) = L</math>

	is '''true''' if the prover has a '''winning strategy''' for this game. The ''winning strategy'' for the prover basically constitutes a strategy to choose an appropriate <math>\delta</math> in terms of the <math>\varepsilon</math> chosen by the skeptic. Thus, it is an expression of <math>\delta</math> as a function of <math>\varepsilon</math>. Verbally, the goal of the prover is to choose a value of <math>\delta</math> so that when the input is restricted to being within <math>\delta</math> distance of <math>x</math>, the output is '''trapped''' to within <math>\varepsilon</math> distance of the claimed limit <math>L</math>.		is '''true''' if the prover has a '''winning strategy''' for this game. The ''winning strategy'' for the prover basically constitutes a strategy to choose an appropriate <math>\delta</math> in terms of the <math>\varepsilon</math> chosen by the skeptic. Thus, it is an expression of <math>\delta</math> as a function of <math>\varepsilon</math>. Verbally, the goal of the prover is to choose a value of <math>\delta</math> so that when the input is restricted to being within <math>\delta</math> distance of <math>c</math>, the output is '''trapped''' to within <math>\varepsilon</math> distance of the claimed limit <math>L</math>.

	We say that the limit statement		We say that the limit statement