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Use the following EXCELLENT interactive Applet to explore Limits at Endpoints:

Additional Tutorials and Practice:

Historical Note About Zeno (Page 1):

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"

Note On Limits of Endpoints (Page 3):

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.)

Practice Questions

1
Question 149 Copy
For the following function, f(x), evaluate the following limits:  ...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
17
Answers
17
Author
Bruce Merz
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19085
2
Question 151
Consider the piecewise function ...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
5
Answers
5
Author
Bruce Merz
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151
3
Question 153
Sketch a possible graph of a function that has domain -3\le x\le 5, has zeros (intersects the x-...
Question Type
Graphing
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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153
4
Question 154
Sketch a possible graph of a function that has domain  x \le 0, but not x=-2,  goes throug...
Question Type
Graphing
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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154
5
Question 155
Sketch a possible graph of a function that has zeros of -2, and 1, and ...
Question Type
Graphing
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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155
6
Question 156
Estimate the value of \mathop{{\rm lim}}_{x\to 0}(1+x)^{\frac{1}{x}} by trying values very close...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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156
7
Question 157
Estimate the value of \mathop{{\rm lim}}_{x\to 2^-}\frac{3x+6}{3x^3-24} by trying values very cl...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
1
Answers
2
Author
Bruce Merz
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157
8
Question 158
For the function f(x)={3x}^2, approximate the slope of the tangent line at (1,3) by doing the fo...
Question Type
Solve for a Variable(s)
Suggested Grade Level
12
Parts
3
Answers
3
Author
Bruce Merz
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158
9
Question 159
For the function f\left(x\right)=-2x^2 , approximate the slope of the tangent line at (1,- 2) us...
Question Type
Solve for a Variable(s)
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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159
10
Question 160
Evaluate the following limit: ...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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160
11
Question 161
Evaluate this limit: ...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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161
12
Question 162
Evaluate the following limit: ...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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162
13
Question 163
Evaluate this limit: ...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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163
14
Question 164
Evaluate this limit: ...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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164
15
Question 165
Evaluate the following limit: ...
Question Type
Evaluate an Expression
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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165
16
Question 3207
AP Prep: Which of the following limits exist? \lim_{x\to -5^{-}}{f(x)} ...
Question Type
Multiple Choice
Suggested Grade Level
12
Parts
15
Answers
15
Author
Bruce Merz
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3207
17
Question 9149
AP Prep: Consider the function: f(x)=\begin{cases}x^2+2 & x<3\\x & x\ge 3\end{cases} ...
Question Type
Multiple Choice
Suggested Grade Level
12
Parts
6
Answers
6
Author
Bruce Merz
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9149
18
Question 3208
[Try This!]: What is the formal definition of a limit of a real function? This is also known as the ...
Question Type
Multi-step Word Problem
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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3208
19
Question 9130
[Try This!]: What is the formal definition of a limit of a real function? This is also known as the ...
Question Type
Proof
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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9130
20
Question 9148
[Try This!]: \lim_{x\to c}f(x)=L is equivalent to: (a) For all \delta, there is an ...
Question Type
Multiple Choice
Suggested Grade Level
12
Parts
1
Answers
1
Author
Bruce Merz
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9148