Determine the following limits. 


lim x→−∞ 40x^4+x^2+5x / √64x^8+x^6

Key Concepts

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. In this context, we analyze how the function behaves as x approaches negative infinity, focusing on the leading terms of the polynomial in both the numerator and denominator, which dominate the behavior of the function.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the given limit, the highest degree terms in the numerator and denominator are crucial for determining the limit's value, as they dictate the function's growth rate as x approaches infinity.

Rational Functions and Simplification

Rational functions are ratios of polynomial functions. To evaluate limits involving rational functions, it is often useful to simplify the expression by dividing the numerator and denominator by the highest power of x present. This simplification helps in identifying the limit more easily, especially when dealing with infinity.