Draw a line at x = 2 and note that the curve approaches this line but does not touch it. Lines or axes that curves approach but never touch are called asymptote lines.
LC02 Graph the function y = 1/x − 2 using limit analysis.
At x = 2, the function has value 1/0, which is hard to interpret. Using the limit concept we can determine how the function behaves as it approaches 2.
First note that the function has the same general features as y = 1/x, so we expect the same general shape but with a shift associated with the x − 2 term.
| When x becomes very large, y
becomes very small. And as x approaches 2 from the positive side,
the denominator of the function becomes a very small positive number,
making y a very large number. As x approaches 2 from the
negative side (numbers slightly less than 2) the denominator becomes a
very small negative number, making y a very large negative
number. As x goes to large negative numbers, y becomes a
very small negative number. Draw a line at x = 2 and note that the curve approaches this line but does not touch it. Lines or axes that curves approach but never touch are called asymptote lines. |
|
Using a well defined limit notation this difference in y as x approaches 2 from the positive side and then the negative side is perhaps a little clearer.
