Understanding the Limit as x Approaches Infinity for Functions with Powers: Comparing Degrees of Powers

lim x->∞ pow top < pow bottom

To find the limit as x approaches infinity of a function with a power on top and a power on the bottom, we need to consider the degrees of the powers

To find the limit as x approaches infinity of a function with a power on top and a power on the bottom, we need to consider the degrees of the powers.

Let’s consider the given expression: lim x->∞ pow top < pow bottom First, let's assume both the power on top and bottom are positive. If the degree of the power on top (let's call it m) is greater than the degree of the power on the bottom (let's call it n), meaning m > n, then as x approaches infinity, the term with the lower power becomes insignificant compared to the term with the higher power. Thus, the value of the expression approaches infinity.

For example, if the expression is given as lim x->∞ x^3 / x^2, as x becomes larger and larger, the term x^2 in the denominator becomes less significant compared to x^3 in the numerator. Therefore, the value of the expression approaches infinity.

On the other hand, if the degree of the power on top is less than the degree of the power on the bottom, meaning m < n, as x approaches infinity, the term with the higher power dominates the expression. In this case, the value of the expression approaches 0 as x approaches infinity. For example, if the expression is given as lim x->∞ x^2 / x^3, as x becomes larger, the term x^3 in the denominator becomes significantly larger than x^2 in the numerator. Thus, the value of the expression approaches 0.

However, if the degrees of the powers on top and bottom are equal, meaning m = n, the limit value cannot be determined without additional information.

It’s important to note that these conclusions apply when we consider positive powers and take the limit as x approaches infinity. Different cases can arise when dealing with negative powers or limits involving other finite values.

In summary, when calculating the limit as x approaches infinity of a function with a power on top and a power on the bottom, compare the degrees of the powers. If the power on top is greater, the limit is infinity. If the power on bottom is greater, the limit is 0.

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