Find each limit, if possible. (a) limx →∞ (3-2 x)/(3 x^3-1) (b) limx →∞ (3-2 x)/(3 x-1) (c) limx

step 1 For part A, we need to figure out the limit as X goes to infinity of the function 3 minus 2x ove

Find each limit, if possible. (a) limx →∞ (3-2 x)/(3 x^3-1)...

Find each limit, if possible.
(a) $\lim _{x \rightarrow \infty} \frac{3-2 x}{3 x^{3}-1}$
(b) $\lim _{x \rightarrow \infty} \frac{3-2 x}{3 x-1}$
(c) $\lim _{x \rightarrow \infty} \frac{3-2 x^{2}}{3 x-1}$

Key Concepts

Degree Comparison of Polynomials

This key idea is based on comparing the degrees of the polynomial expressions in the numerator and denominator. When the degree of the numerator is less than that of the denominator, the limit is zero; if they are equal, the limit is the ratio of the leading coefficients; and if the degree of the numerator exceeds that of the denominator, the limit may be infinite or not exist. This comparison is central to efficiently evaluating limits at infinity of rational functions.

Algebraic Simplification by Dividing Through by the Highest Power of x

To rigorously evaluate limits, one common technique is to divide both the numerator and the denominator by the highest power of x present in either. This process simplifies the expression by reducing lower degree terms to negligible fractions, making it clear which parts of the function dominate as x approaches infinity. This method not only simplifies the evaluation process but also reinforces the role of the dominant terms in determining the limit.

Limits of Rational Functions at Infinity

This concept involves evaluating the behavior of a ratio of polynomial functions as the variable approaches infinity. The method relies on understanding that as x grows larger, the highest degree terms in the numerator and denominator dominate the behavior of the function, while lower degree terms become negligible. This concept is crucial for determining whether the limit tends toward zero, a finite nonzero value, or infinity.

Transcript

Please give Ace some feedback