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Verbal Description of Law |
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When x is approaching the value of c, there is no effect whatsoever on b. |
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This limit is evaluated by using the definition of the 
notation . |
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If an exponential expression has its base approaching a limit, the exponential expression will approach the limit to the same power. |
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The root of a number which is approaching a limit is the root of the limit. |
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The limit of a constant times a function is the constant times the limit of the function. |
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The limit of a sum is the sum of the limits, and
The limit of a difference is the difference of the limits. |
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The limit of a product is the product of the limits. |
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The limit of a quotient is the quotient of the limits, if the limit in the denominator is not equal to zero. |
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The limit of a function to a power can be calculated by taking the power of the function and then taking the limit, or by taking the limit and then raising the limit to the same power. |
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This is really identical to the property immediately above, with the power being a fraction. |
| # | Technique for calculating:  |
| 1 | Try substituting a into the limit expression. If you can solve this expression, you're done. This technique only works when the function is continuous at x = a. |
| 2 | If after you substitute, you can't simplify, try simplifying algebraically first, then substitute. |
| 3 | If you get a number for the numerator and a zero in the denominator, then there is no limit. |
| 4 | If you get zero in the numerator and the denominator, keep working, there is a limit. |
| 5 | To evaluate the limit of a rational function at infinity, divide numerator and denominator by the highest power of x that shows up in the denominator. Then subsitute to find the limit. |
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Given example problem. |
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When substituting, we get 0/0 which is of course impossible, and yields no information about the limit we are trying to calculate. The 0/0 result from substituting tells us that there must be a limit, therefore we.... |
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We try technique #2, and we first attempt to algebraically simplify. Here we factor the numerator and cancel. |
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Now that we've simplified, we try substitution again. This works, and we are done. |
Example Problem:
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Given example problem. |
| Direct substitution of 0 for t fails. | When substituting, we get 0/0 which is of course impossible, and yields no information about the limit we are trying to calculate. The 0/0 result from substituting tells us that there must be a limit, therefore we.... |
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We then work on simplifying algebraically. Here multiplying the numerator and denominator by the conjugate of the numerator works well. |
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Simplifying after this multiplication gets us to this point. |
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And now substitution works just fine. |
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If you have the Journey Through Calculus CD, load and run MResources/Module 2/The Essential Examples/Examples D and E. This module will allow you to explore limits interactively. |
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If you have the Journey Through Calculus CD, load and run MResources/Module 2/The Essential Examples/Example C . This module will allow you to explore limits interactively. |
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If you have the Journey Through Calculus CD, load and run MResources/Module 2/Basics of Limits/Sound of a Limit that Exists. This module will allow you to watch an animation of a limit. |
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