StudyForge Samples > Chapter 11: Calculus Samples > Lesson 1: Introduction to Limits

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- An introduction to limits from an intuitive point of view

http://archives.math.utk.edu/visual.calculus/1/limits.16/index.html

The above paradox was originally proposed by Zeno of Alea in a more general form, in the sense that continually halfing a distance will result in never achieving the whole. It has been expressed through various means, including races such as the above, and was also used to explain the inability of the warrior Achilles to catch a turtle: if the warrior advances even slightly, so will the turtle, and thus he will never catch up. "And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount." - Simplicious, from "On Aristotle's Physics"

Some resources/professors will say that for an endpoint of the domain (such as x=0, 5 in the Ex. on Page 3), that because you don't have both a left and a right-hand limit, that the limit doesn't exist; other resources or professors will say that it does exist, and is simply equal to the one sided limit there. The problem is that either way, it's actually a point of mathematical inconsistency in Calculus, since a special allowance for endpoints will have to made at some point, based on current definitions. Those who adopt the former definition have to make these special allowances for endpoints when it comes to the upcoming topic of Continuity, not to mention Differentiability afterwards, whereas those who adopt the latter convention, as we have, make the special allowance only at Limits, and then it's consistent from that point on, not to mention more intuitive. (We won't ask any questions on that specific idea here though, to respect the differences.)

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Question 149 Copy

For the following function, , evaluate the following limits: ...

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Question 151

Consider the piecewise function ...

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Question 153

Sketch a possible graph of a function that has domain , has zeros (intersects the x-...

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Question 154

Sketch a possible graph of a function that has domain , but not , goes throug...

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Question 155

Sketch a possible graph of a function that has zeros of -2, and 1, and ...

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Question 156

Estimate the value of by trying values very close...

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Question 157

Estimate the value of by trying values very cl...

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Question 158

For the function , approximate the slope of the tangent line at (1,3) by doing the fo...

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Question 159

For the function , approximate the slope of the tangent line at (1,- 2) us...

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Question 160

Evaluate the following limit: ...

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Question 161

Evaluate this limit: ...

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Question 162

Evaluate the following limit: ...

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Question 163

Evaluate this limit: ...

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Question 164

Evaluate this limit: ...

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Question 165

Evaluate the following limit: ...

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Question 3207

AP Prep: Which of the following limits exist? ...

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Question 9149

AP Prep: Consider the function: ...

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Question 3208

[Try This!]: What is the formal definition of a limit of a real function? This is also known as the ...

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Question 9130

[Try This!]: What is the formal definition of a limit of a real function? This is also known as the ...

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Question 9148

[Try This!]: is equivalent to: (a) For all , there is an ...

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