Let's study the function . To find the end behavior of this function, evaluate the limit of g(x) as x approaches infinity.

If we divide each term by x, we get the equivalent expression . As x becomes large, approaches zero. So becomes large. Similarly, as x becomes large, approaches 0. So gets close to -1.

So, as x becomes large, the numerator becomes large. However, the denominator gets closer to -1. Dividing a large number by -1, results in a large negative number. This means that the limit as x approaches infinity is negative infinity.

Look at the graph of this function. You can see that as x approaches infinity, g(x) approaches negative infinity, and as x approaches negative infinity, g(x) approaches infinity. In fact, as x becomes large, g(x) behaves similar to a linear function. We say that g(x) has an oblique asymptote — a linear asymptote that is neither vertical nor horizontal.

You saw that for very large values of x, the behavior of the function approaches the behavior of a linear function.

Let's see how to find the oblique asymptote of g(x) algebraically. To do so, divide 3 - x into x2 - 2x + 1. Look to the right to see how this division is performed.

So, g(x) can be written as the sum of -x - 1 and .

As x approaches positive or negative infinity, the remainder approaches zero and therefore g(x) approaches -x - 1. So, the line y = -x - 1 is the oblique asymptote of this function.

If the degree of the numerator of a rational function is one greater than the degree of the denominator, then the graph has an oblique asymptote.

Decide whether the function given has a horizontal asymptote or an oblique asymptote. Then find this asymptote. Click "Submit" when finished.

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