Limits and Infinity


Find the limits in Exercises 37–46.


2x² + 3
lim -------------
x→⁻∞ 5x² + 7

Verified step by step guidance

1. Identify the highest degree terms in the numerator and the denominator. In this case, both the numerator and the denominator have the highest degree term of x².
2. Rewrite the limit expression by factoring out the highest degree term from both the numerator and the denominator. This gives us: lim (x→⁻∞) [(2x²/x²) + (3/x²)] / [(5x²/x²) + (7/x²)].
3. Simplify the expression by canceling out the x² terms in the numerator and the denominator. This results in: lim (x→⁻∞) [2 + (3/x²)] / [5 + (7/x²)].
4. Evaluate the limit as x approaches negative infinity. As x becomes very large in magnitude, the terms (3/x²) and (7/x²) approach zero.
5. Conclude that the limit is the ratio of the leading coefficients of the highest degree terms, which is 2/5.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions at specific points, including points of discontinuity or infinity. In this context, we are interested in the limit of a rational function as x approaches negative infinity.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They can exhibit different behaviors as the variable approaches certain values, including infinity. Analyzing the leading terms of the numerator and denominator is crucial for determining the limit of a rational function at infinity.

Behavior at Infinity

The behavior of functions as they approach infinity is essential for evaluating limits. For rational functions, this often involves comparing the degrees of the polynomials in the numerator and denominator. If the degrees are the same, the limit is the ratio of the leading coefficients; if the degree of the numerator is less, the limit approaches zero.