Analyzing the Limit of a Function as x Approaches Infinity: A Guide to the Highest-Power Terms

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To find the limit of a function as x approaches infinity, we need to analyze the ratio of the highest-power terms in the numerator and denominator

To find the limit of a function as x approaches infinity, we need to analyze the ratio of the highest-power terms in the numerator and denominator.

If we have a function with a polynomial numerator and a polynomial denominator, we can determine the limit as x approaches infinity by examining the ratio of the highest-power terms in the numerator and denominator.

Let’s consider the function:

lim x->∞ (ax^n + bx^(n-1) + … + c) / (dx^m + ex^(m-1) + … + f)

In this expression, ax^n is the highest-power term in the numerator and dx^m is the highest-power term in the denominator.

To evaluate the limit, we need to compare the exponents n and m:

1. If n > m, then the highest-power term in the numerator dominates the expression, and the limit is positive or negative infinity, depending on the signs of a and d.

2. If n = m, then both the numerator and denominator have the same highest-power term. In this case, we can divide each term by x^m, which simplifies the expression:

lim x->∞ [(ax^n + bx^(n-1) + … + c) / x^m ] / [(dx^m + ex^(m-1) + … + f) / x^m ]

Taking the limit of this simplified expression as x approaches infinity, we get:

lim x->∞ (a + b/x + … + c/x^n) / (d + e/x + … + f/x^m)

Since the highest-power terms have been eliminated, we can proceed to analyze the limit using the other terms. If all the coefficients are finite, the limit is a finite number.

3. If n < m, then the highest-power term in the denominator dominates the expression, and the limit is 0. Dividing both the numerator and denominator by x^m, the expression becomes: lim x->∞ [(ax^n + bx^(n-1) + … + c) / x^m ] / [(dx^m + ex^(m-1) + … + f) / x^m ]

As x approaches infinity, all the terms in the numerator converge to zero, while the terms in the denominator remain finite. Therefore, the limit is 0.

In some cases, the highest-power terms in the numerator and denominator might cancel out, resulting in a finite limit. However, this is not the general case and further analysis is required to determine the exact limit value.

It is important to note that these rules are specific to polynomial functions and may not apply in other cases involving exponential, logarithmic, or trigonometric functions.

In summary, to find the limit of a function as x approaches infinity, we compare the highest-power terms in the numerator and denominator. If the exponent in the numerator is greater, the limit is positive or negative infinity. If the exponents are equal, the limit could be finite or infinite, depending on the coefficients. If the exponent in the denominator is greater, the limit is 0.

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