Understanding the Limit of a Fraction with Numerator and Denominator Tending to Infinity

lim x->∞ pow top > pow bottom

Looking for the limit of a fraction where the numerator and denominator both tend to infinity

Looking for the limit of a fraction where the numerator and denominator both tend to infinity. Let’s break down the problem step by step.

Suppose we have a function f(x) = top(x) / bottom(x), where top(x) and bottom(x) represent two functions that both tend to infinity as x approaches infinity.

To find the limit as x tends to infinity, we can analyze the behavior of the function as x becomes very large.

One approach is to look at the rate at which the numerator and denominator grow. If the numerator grows faster than the denominator, the overall function will also tend to infinity. If the denominator grows faster, the function tends to zero.

To determine which function grows faster, we can consider the highest power of x in both the numerator and denominator.

Let’s say the highest power of x in the numerator is n and in the denominator is m. If n < m, then as x tends to infinity, the denominator will dominate the numerator, and the function will tend to zero. If n > m, then the numerator will dominate, and the function will tend to infinity. If n = m, then we need to consider the coefficients of the leading terms to determine the behavior of the function.

For example, let’s say we have the function f(x) = x^2 / x^3 as x tends to infinity. The highest power of x in the numerator is 2, and in the denominator, it is 3. Since 2 < 3, the denominator grows faster, and as x approaches infinity, the function approaches zero. On the other hand, if we have f(x) = x^3 / x^2 as x tends to infinity, the highest power of x in the numerator is 3, while in the denominator, it is 2. Since 3 > 2, the numerator grows faster, and as x approaches infinity, the function approaches infinity.

In the case where n = m, let’s take an example: f(x) = 2x^3 / 3x^3 as x tends to infinity. Here, both the numerator and denominator have the same highest power, which is 3. Since the coefficients of the leading terms (2 and 3 in this case) are different, we can see that as x approaches infinity, the function will tend to the ratio of the coefficients, which is 2/3. So, in this case, the limit of f(x) as x tends to infinity would be 2/3.

To summarize, when you have a fraction with both the numerator and denominator tending to infinity, you can determine the limit by comparing the highest powers of x. If the power in the numerator is greater, the limit is ∞. If the power in the denominator is greater, the limit is 0. If the powers are equal, look at the leading coefficients to determine the limit.

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