Limits and Infinity


Find the limits in Exercises 37–46.


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

Verified step by step guidance

1. Identify the type of limit: As \( x \to -\infty \), we are dealing with a rational function limit where both the numerator and the denominator are polynomials.
2. Compare the degrees of the polynomials: The degree of the numerator \( 2x^2 + 3 \) is 2, and the degree of the denominator \( 5x^2 + 7 \) is also 2.
3. Since the degrees of the numerator and the denominator are the same, the limit as \( x \to -\infty \) is determined by the ratio of the leading coefficients.
4. Identify the leading coefficients: The leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 5.
5. Calculate the limit by taking the ratio of the leading coefficients: \( \lim_{{x \to -\infty}} \frac{2x^2 + 3}{5x^2 + 7} = \frac{2}{5} \).