Asymptotic Behavior of Fractions with Power Functions: Evaluating the Limit as x Approaches Infinity

lim x->∞ pow top = pow bottom

To find the limit of a fraction as x approaches infinity with a power in the numerator and denominator, we can use the concept of asymptotes

To find the limit of a fraction as x approaches infinity with a power in the numerator and denominator, we can use the concept of asymptotes.

Let’s consider the expression:

lim x->∞ (top^a) / (bottom^b)

where a and b are positive integers, and top and bottom represent functions of x.

To evaluate this limit, we need to understand how the numerator and denominator behave as x approaches infinity. There are a few scenarios to consider:

1. If a > b:
In this case, the numerator grows faster than the denominator. As x gets larger, the term (top^a) will dominate, and the fraction will tend to positive or negative infinity. We can conclude that the limit does not exist.

2. If a < b: In this case, the denominator grows faster than the numerator. As x approaches infinity, the term (bottom^b) will dominate, and the fraction will tend to 0. Thus, the limit is 0. 3. If a = b: When the powers in the numerator and denominator are equal, we can simplify the expression to: lim x->∞ (top^a) / (bottom^a)

In this situation, the numerator and denominator have the same growth rate. We can divide both the numerator and denominator by the highest power of x in the expression, which will lead to a simplified form. For example, if top and bottom are polynomial functions of x, we divide each term by x^a. After simplifying, we can evaluate the limit based on the coefficients and exponents.

In summary, the limit of a fraction with a power in the numerator and denominator as x approaches infinity depends on the relative growth rates of the two terms. If the numerator grows faster, the limit diverges; if the denominator grows faster, the limit is 0; and if the powers in the numerator and denominator are equal, further simplification may be required to find the limit.

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