1. Explain the difference between a value at and the limit as approaches .
2. Explain why we say a function does not have a limit as approaches if, as approaches , the left-hand limit is not equal to the right-hand limit.
For the following exercises, estimate the functional values and the limits from the graph of the function provided in Figure 14.

Figure 14
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For the following exercises, draw the graph of a function from the functional values and limits provided.
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For the following exercises, use a graphing calculator to determine the limit to 5 decimal places as approaches 0.
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27. Based on the pattern you observed in the exercises above, make a conjecture as to the limit of , , .
For the following exercises, use a graphing utility to find graphical evidence to determine the left- and right-hand limits of the function given as approaches . If the function has a limit as approaches , state it. If not, discuss why there is no limit.
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For the following exercises, use numerical evidence to determine whether the limit exists at . If not, describe the behavior of the graph of the function near . Round answers to two decimal places.
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For the following exercises, use a calculator to estimate the limit by preparing a table of values. If there is no limit, describe the behavior of the function as approaches the given value.
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For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and right-hand limits of the function given as approaches . If the function has a limit as approaches , state it. If not, discuss why there is no limit.
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50. Use numerical and graphical evidence to compare and contrast the limits of two functions whose formulas appear similar: and as approaches 0. Use a graphing utility, if possible, to determine the left- and right-hand limits of the functions and as approaches 0. If the functions have a limit as approaches 0, state it. If not, discuss why there is no limit.
51. According to the Theory of Relativity, the mass of a particle depends on its velocity . That is
where is the mass when the particle is at rest and is the speed of light. Find the limit of the mass, , as approaches .
52. Allow the speed of light, , to be equal to 1.0. If the mass, , is 1, what occurs to as Using the values listed in the table below, make a conjecture as to what the mass is as approaches 1.00.
0.5 | 1.15 |
0.9 | 2.29 |
0.95 | 3.20 |
0.99 | 7.09 |
0.999 | 22.36 |
0.99999 | 223.61 |
Candela Citations
- Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface. License: CC BY: Attribution. License Terms: Download for free at: http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface