Understanding Limits: Evaluating the Limit as x Approaches Infinity of a Power Function with Higher Exponent Numerator

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To evaluate the limit as x approaches infinity of a power function where the exponent of the numerator is greater than the exponent of the denominator, we need to consider the leading terms of both parts of the function

To evaluate the limit as x approaches infinity of a power function where the exponent of the numerator is greater than the exponent of the denominator, we need to consider the leading terms of both parts of the function.

Let’s assume we have the function f(x) = (ax^m)/(bx^n).

If the exponent m is greater than n, we can ignore the lower-degree terms as their contribution becomes negligible compared to the leading terms as x grows larger.

To find the limit as x approaches infinity, we focus on the term with the highest degree, which is ax^m in the numerator. In this case, as x becomes infinitely large, the dominant term with the highest degree grows without bound.

Hence, when the exponent of the numerator is greater than the exponent of the denominator (m > n), the limit as x approaches infinity is positive infinity (∞).

Mathematically, we can represent this as:

lim x->∞ (ax^m)/(bx^n) = ∞

Note that this assumes the ratio (a/b) is not zero or undefined, and the exponents m and n are finite and positive.

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