Understanding Limits | Comparing Powers as x Approaches Infinity

lim x->∞ pow top < pow bottom

The expression “lim x->∞ pow top < pow bottom" seems to be incomplete, as it is missing the actual mathematical expressions for "pow top" and "pow bottom

The expression “lim x->∞ pow top < pow bottom" seems to be incomplete, as it is missing the actual mathematical expressions for "pow top" and "pow bottom." However, based on the context, I can provide an explanation for the general concept of limits and how they relate to the comparison between two powers. In mathematics, a limit represents the value that a function approaches as the variable (in this case, x) gets arbitrarily close to a certain point (in this case, positive infinity). We use the notation "lim x->∞” to indicate that we are considering the limit as x tends towards infinity.

When you mention “pow top” and “pow bottom,” I assume you mean two expressions involving powers of x, such as x^m and x^n, where m and n are real numbers. In this case, the comparison between the two powers can influence the behavior of the limit.

To determine whether pow top is less than pow bottom in the limit as x approaches infinity, we need to consider the values of m and n. Here are some possibilities:

1. If m > n:
In this case, as x becomes larger, the contribution of x^m will dominate over x^n. Therefore, pow top will ultimately become larger than pow bottom. Mathematically, we can write:
lim x->∞ x^m > x^n

2. If m < n: Similarly, if m is smaller than n, then x^n will have a greater influence as x tends towards infinity, and pow top would be less than pow bottom. Formally: lim x->∞ x^m < x^n 3. If m = n: When m and n are equal, the powers of x have the same weight, and the result depends on the coefficients or other terms in the expressions. Comparing pow top and pow bottom would require additional information to determine their relationship. To summarize, when dealing with limits as x approaches infinity and comparing two powers, the answer will depend on the relative sizes of the exponents.

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