Question

if the largest exponent in the numerator is larger than the largest exponent in the denominator then.....

Answer 1

If the largest exponent in the numerator is larger than the largest exponent in the denominator then the expression diverges to infinity.

Step-by-step explanation:

In mathematical terms, when the largest exponent in the numerator (let's denote it as n₁) is greater than the largest exponent in the denominator (denoted as n₂), the overall fraction becomes unbounded and tends towards infinity. This occurs because, as the variable in the expression approaches infinity, the term with the larger exponent in the numerator dominates the fraction.

Consider the expression
`\( \frac{{a_n x^(n₁) + a_(n-1) x^(n₁-1) + \ldots + a_1 x + a_0}}{{b_m x^(n₂) + b_(m-1) x^(n₂-1) + \ldots + b_1 x + b_0}} \) where \( n₁ > n₂ \)`.

As x becomes larger, the terms
`\( a_n x^(n₁) \) and \( b_m x^(n₂) \)` will have the most significant impact on the overall value of the expression. Since
`\( n₁ > n₂ \)`, the numerator's leading term
`\( a_n x^(n₁) \)` will grow faster than the denominator's leading term
`\( b_m x^(n₂) \)`, causing the entire fraction to grow without bound.

This divergence to infinity is a fundamental concept in calculus and is often encountered when analyzing the behavior of functions or sequences. It signifies that as the input values become extremely large, the function or sequence represented by the expression will also become arbitrarily large, indicating an unbounded trend in the mathematical model.