Quiz:Limit - Revision history

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

2012-09-29T19:10:51Z

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

← Older revision		Revision as of 19:10, 29 September 2012
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	+ Every smaller positive value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.		+ Every smaller positive value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.
	\|\| See from the definition or the game description.		\|\| See from the definition or the game description.
	- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of $\delta$ works for every smaller positive value of <math>\varepsilon</math>.		- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every smaller positive value of <math>\varepsilon</math>.
	- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.		- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.
	- None of the above statements need always be true.		- None of the above statements need always be true.

Vipul: /* Left hand limit and right hand limit */

2012-09-29T19:09:20Z

Left hand limit and right hand limit

← Older revision		Revision as of 19:09, 29 September 2012
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	- The left hand limit at any point other than the left endpoint and the right hand limit at any point other than the right endpoint.		- The left hand limit at any point other than the left endpoint and the right hand limit at any point other than the right endpoint.

			{Suppose <math>c</math> is a real number and <math>f</math> is a function whose domain contains an open interval of the form <math>(c - t,c)</math> for some <math>t > 0</math>. Which of the following is true?
			\|type="()"}
			- <math>\lim_{x \to c^-} f(x)</math> exists if and only if <math>f</math> is continuous on the immediate left of <math>c</math>, i.e., there exists <math>\delta > 0</math> such that <math>f</math> is continuous on <math>(c - \delta, c)</math>. Here, "continuous" means that the two-sided limit at every point in the open interval equals the value of the function.
			- <math>\lim_{x \to c^-} f(x)</math> exists implies that <math>f</math> is continuous on the immediate left of <math>c</math>, i.e., there exists <math>\delta > 0</math> such that <math>f</math> is continuous on <math>(c - \delta, c)</math>. Here, "continuous" means that the two-sided limit at every point in the open interval equals the value of the function. However, the converse implication does not hold.
			- <math>\lim_{x \to c^-} f(x)</math> exists if <math>f</math> is continuous on the immediate left of <math>c</math>, i.e., there exists <math>\delta > 0</math> such that <math>f</math> is continuous on <math>(c - \delta, c)</math>. Here, "continuous" means that the two-sided limit at every point in the open interval equals the value of the function. However, the converse implication does not hold.
			+ Neither of the conditions "<math>\lim_{x \to c^-} f(x)</math> exists" and "<math>f</math> is continuous on the immediate left of <math>c</math>" imply one another.
			\|\| We can construct examples illustrating both phenomena. A function that has one piece definition for rationals with another piece definition for irrationals, and the two definitions having the same limit at <math>c</math>, would illustrate how the left hand limit at <math>c</math> can exist without the function being continuous at <math>c</math>. For instance, <math>f(x) := \left \lbrace \begin{array}{rl} 0, & x \mbox{ irrational } \\x, & \mbox{ rational } \\\end{array}\right.</math> at <math>c = 0</math>. For the other direction, consider a function like <math>f(x) := 1/x, x \ne 0</math> at <math>c = 0</math>. This is continuous on the left of 0 but does not have a left hand limit at 0.
	</quiz>		</quiz>

Vipul at 19:00, 29 September 2012

2012-09-29T19:00:35Z

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 {{perspectives}}
 ==Definition for finite limit for finite function of one variable==
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 <quiz display=simple>
-{Suppose <math>c,L \in \R</math> and <math>f</math> is a function defined on a subset of <math>\R</math>. Which of these is the correct interpretation of <math>\lim_{x \to c} f(x) = L</math> in terms of the definition of limit?
+{Suppose <math>c,L \in \R</math> and <math>f</math> is a function defined on a subset of <math>\R</math>. Which of these is the correct interpretation of <math>\displaystyle \lim_{x \to c} f(x) = L</math> in terms of the definition of limit?
 |type="()"}
 - For every <math>\alpha > 0</math>, there exists <math>\beta > 0</math> such that if <math>0 < |x - c| < \alpha</math>, then <math>|f(x) - L| < \beta</math>.
@@ Line 25: / Line 53: @@
 </quiz>
 ===Left hand limit and right hand limit===

Vipul: /* Left hand limit and right hand limit */

2012-09-28T00:59:43Z

Left hand limit and right hand limit

← Older revision		Revision as of 00:59, 28 September 2012
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	+ For every <math>\varepsilon > 0</math>, there exists <math>\delta > 0</math> such that for all <math>x \in \R</math> satisfying <math>0 < c - x < \delta</math>, we have <math>\|f(x) - L\| < \varepsilon</math>.		+ For every <math>\varepsilon > 0</math>, there exists <math>\delta > 0</math> such that for all <math>x \in \R</math> satisfying <math>0 < c - x < \delta</math>, we have <math>\|f(x) - L\| < \varepsilon</math>.

	{Suppose the domain of a function <math>f</math> is a closed bounded interval (i.e., an interval of the form <math>[a,b]</math> for real numbers <math>a,b</math>. Which of the following definitely ''do '''not''' make sense''?		{Suppose the domain of a function <math>f</math> is a closed bounded interval, i.e., an interval of the form <math>[a,b]</math> for real numbers <math>a,b</math>. Which of the following definitely ''do '''not''' make sense''?
	\|type="()"}		\|type="()"}
	+ The left hand limit at the left endpoint and the right hand limit at the right endpoint.		+ The left hand limit at the left endpoint and the right hand limit at the right endpoint.

Vipul: /* Non-existence of limit */

2012-09-07T22:09:18Z

Non-existence of limit

← Older revision		Revision as of 22:09, 7 September 2012
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	==Non-existence of limit==		==Non-existence of limit==

			<quiz display=simple>
	{Suppose <math>f</math> is a function defined on a subset of <math>\R</math>. <math>c</math> is a real number such that <math>(c - t,c) \cup (c, c + t)</math> is in the domain of <math>f</math> for some <math>t > 0</math>. Identify the correct interpretation of the statement "<math>\lim_{x \to c} f(x)</math> does not exist" among the choices below.		{Suppose <math>f</math> is a function defined on a subset of <math>\R</math>. <math>c</math> is a real number such that <math>(c - t,c) \cup (c, c + t)</math> is in the domain of <math>f</math> for some <math>t > 0</math>. Identify the correct interpretation of the statement "<math>\lim_{x \to c} f(x)</math> does not exist" among the choices below.
	\|type="()"}		\|type="()"}

Vipul at 22:06, 7 September 2012

2012-09-07T22:06:17Z

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 - Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.
 - None of the above statements need always be true.
 </quiz>

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

2012-09-07T21:59:13Z

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

← Older revision		Revision as of 21:59, 7 September 2012
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	- Every smaller positive value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every smaller positive value of <math>\varepsilon</math>.		- Every smaller positive value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every smaller positive value of <math>\varepsilon</math>.
	+ Every smaller positive value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.		+ Every smaller positive value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.
			\|\| See from the definition or the game description.
	- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of $\delta$ works for every smaller positive value of <math>\varepsilon</math>.		- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of $\delta$ works for every smaller positive value of <math>\varepsilon</math>.
	- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.		- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.

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

2012-09-07T21:58:35Z

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

← Older revision		Revision as of 21:58, 7 September 2012
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	<quiz display=simple>		<quiz display=simple>
	{In the usual <math>\~~epsilon~~-\delta</math> definition of limit for a given limit <math>\lim_{x \to c} f(x) = L</math>, if a given value <math>\delta > 0</math> works for a given value <math>\~~epsilon~~ > 0</math>, then which of the following is true?		{In the usual <math>\varepsilon-\delta</math> definition of limit for a given limit <math>\lim_{x \to c} f(x) = L</math>, if a given value <math>\delta > 0</math> works for a given value <math>\varepsilon > 0</math>, then which of the following is true?
	\|type="()"}		\|type="()"}
	- Every smaller positive value of <math>\delta</math> works for the same <math>\~~epsilon~~</math>. Also, the given value of <math>\delta</math> works for every smaller positive value of <math>\~~epsilon~~</math>.		- Every smaller positive value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every smaller positive value of <math>\varepsilon</math>.
	+ Every smaller positive value of <math>\delta</math> works for the same <math>\~~epsilon~~</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\~~epsilon~~</math>.		+ Every smaller positive value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.
	- Every larger value of <math>\delta</math> works for the same <math>\~~epsilon~~</math>. Also, the given value of $\delta$ works for every smaller positive value of <math>\~~epsilon~~</math>.		- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of $\delta$ works for every smaller positive value of <math>\varepsilon</math>.
	- Every larger value of <math>\delta</math> works for the same <math>\~~epsilon~~</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\~~epsilon~~</math>.		- Every larger value of <math>\delta</math> works for the same <math>\varepsilon</math>. Also, the given value of <math>\delta</math> works for every larger value of <math>\varepsilon</math>.
	- None of the above statements need always be true.		- None of the above statements need always be true.

	</quiz>		</quiz>

Vipul at 21:57, 7 September 2012

2012-09-07T21:57:35Z

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 - The two-sided limit at any interior point.
 - The left hand limit at any point other than the left endpoint and the right hand limit at any point other than the right endpoint.
 </quiz>

Vipul: /* Left hand limit and right hand limit */

2012-09-07T21:52:13Z

Left hand limit and right hand limit

← Older revision		Revision as of 21:52, 7 September 2012
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	<quiz display=simple>		<quiz display=simple>
	{Which of these is the correct interpretation of <math>\lim_{x \to c} f(x) = L</math> in terms of the definition of limit?		{Which of these is the correct interpretation of the left hand limit <math>\lim_{x \to c^-} f(x) = L</math> in terms of the definition of limit?
	\|type="()"}		\|type="()"}
	- For every <math>\varepsilon > 0</math>, there exists <math>\delta > 0</math> such that for all <math>x \in \R</math> satisfying <math>0 < c - x < \delta</math>, we have <math>0 < L - f(x) < \varepsilon</math>.		- For every <math>\varepsilon > 0</math>, there exists <math>\delta > 0</math> such that for all <math>x \in \R</math> satisfying <math>0 < c - x < \delta</math>, we have <math>0 < L - f(x) < \varepsilon</math>.