Find each limit, if it exists. limx →-∞ (-3 x^2+5)/(2-x) |...

Name: Problem 72, Chapter 2: Calculus and Its Applications
Uploaded: 2020-10-23T17:44:11-08:00
Duration: 2 min 10 s
Channel: Nick Johnson
Description: Find each limit, if it exists. limx →-∞ (-3 x^2+5)/(2-x)

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Key Concepts

Dominant Term Analysis

When evaluating limits of rational functions as x tends to infinity or negative infinity, the dominant term in both the numerator and the denominator (i.e., the term with the highest power of x) controls the behavior of the function. In such analyses, less significant terms become negligible in comparison, allowing for simpler estimation of the limit.

Degree Comparison in Rational Functions

The behavior of a rational function at infinity depends largely on the degrees of the polynomials in the numerator and the denominator. If the numerator's degree is higher than the denominator's, the magnitude of the function typically increases without bound, whereas if it is lower, the function tends to zero. When the degrees are equal, the limit is determined by the ratio of the leading coefficients.

Techniques for Evaluating Limits at Infinity

A common approach to finding limits at infinity is to factor out the highest power of x from both the numerator and denominator, which enables direct comparison of the dominant terms. This technique simplifies the expression and leads to determination of the limit based on how these leading terms behave as x approaches positive or negative infinity.

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