The values of the function (3 x^{2} + 1)/x^{2} get very close to 3 for very large and very large negative values of x. (Go back to "Limits and Asymptotes" to see a graph.)
A function such as the inverse tangent function w = tan^{-1}(z) gets close to one value
The exponential function e^{x} gets very close to zero for even moderately large negative values and is very big for large positive values.
Similarly, even degree polynomial functions such as x^{4} - 100 x - 10000 grow very large for large positive and negative values of x, while odd degree polynomial functions such as x^{5} - 100 x - 10000 grow large for large values of x and very negative for large negative values of x.
All of this behavior can be stated in terms of limits at and at - as follows:
3 x^{2} + 1 | 3 x^{2} + 1 | ||||
lim | = 3 | lim | = 3 | ||
x --> | x^{2} | x --> - | x^{2} |
lim | arctan(z) | = /2 | lim | arctan(z) | = -/2 |
z --> | z --> - |
lim | e^{x} | = | lim | e^{x} | = 0 |
x --> | x --> - |
lim | (x^{4} - 100 x - 10000) | = | lim | (x^{4} - 100 x - 10000) | = |
x --> | x --> - |
lim | (x^{5} - 100 x - 10000) | = | lim | (x^{5} - 100 x - 10000) | = - |
x --> | x --> - |
After all we have been through, understanding the ten limits above should be a stroll through the woods, but if any logs are blocking the path, there are more examples in the practice area.
Finally, we do not have a formal definition of a limit at or -. If we try to substitute the symbol for the objective a in the definition of limit directly, we come up with the phrase
which doesn't make a whole lot of sense without translation. What is the correct translation? You might want to think about this yourself before looking at the official definition in the Glossary. After that, try the following equipment check.
Equipment Check: Take out a pencil and some scratch paper and make a simple sketch of a function f(x) all of whose vertical and horizontal asymptotes are expressed by the following limits:
lim | f(x) | = 7 | lim | f(x) | = | lim | f(x) | = - |
x --> | x --> - | x --> -2 |
lim | f(x) | = | lim | f(x) | = - |
x -->3^{-} | x -->3 ^{+} |
Click on the button to see a sample answer.
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CalculusQuest^{TM}
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson