Determine the following limits at infinity.


lim x→−∞ 3x^11

Verified step by step guidance

1. Identify the function whose limit we need to find: \( f(x) = 3x^{11} \).
2. Recognize that as \( x \to -\infty \), the term \( x^{11} \) will dominate the behavior of the function because it is a polynomial.
3. Since the exponent 11 is odd, \( x^{11} \) will be negative when \( x \) is negative.
4. Multiply by the constant 3, which will not change the sign of the expression, so \( 3x^{11} \) will also be negative as \( x \to -\infty \).
5. Conclude that the limit of \( 3x^{11} \) as \( x \to -\infty \) is \(-\infty\).

Key Concepts

Limits at Infinity

Limits at infinity refer to the behavior of a function as the input approaches positive or negative infinity. This concept helps in understanding how functions behave for very large or very small values of x, which is crucial for analyzing polynomial, rational, and other types of functions.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The degree of the polynomial, which is the highest power of the variable, significantly influences its end behavior as x approaches infinity or negative infinity.

End Behavior of Functions

End behavior describes how a function behaves as the input values become very large or very small. For polynomial functions, the leading term (the term with the highest degree) dominates the function's behavior at infinity, determining whether the limit approaches positive or negative infinity.