Question

Determine the following limits at infinity. $$\lim _{x \rightarrow-\infty} 3 x^{11}$$

Short answer

Question: Find the limit of the function 3x^11 as x approaches negative infinity. Answer: The limit of 3x^11 as x approaches negative infinity is -∞.

Step-by-step solution

Identifying the degree and leading coefficient

The given polynomial is 3x^11. The degree of the polynomial is 11, which is an odd degree. The leading coefficient is 3.

Evaluating limit at negative infinity

As x approaches negative infinity, we want to determine the behaviour of 3x^11. Since the degree is odd and the leading coefficient is positive, the function will tend to negative infinity as x approaches negative infinity. This can be written as: $$\lim _{x \rightarrow-\infty} 3x^{11} = -\infty$$

Key concepts

Polynomial Functions

Polynomial functions are algebraic expressions composed of terms with non-negative integer exponents and real coefficients. A simple example is the quadratic function f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. In general, a polynomial of degree n can be written as f(x) = a_nx^n + a_{n−1}x^{n−1} + ... + a_2x^2 + a_1x + a_0, where a_n, a_{n−1}, ..., a_2, a_1, and a_0 are real numbers, and a_n ≠ 0.

The behavior of a polynomial function as x approaches infinity is largely driven by its term with the highest power, as this term will grow faster than the others as x increases or decreases. The exercise given involves such an evaluation, focusing on the behavior as x tends towards negative infinity.

Leading Coefficient

The leading coefficient of a polynomial function is the coefficient of the term with the highest degree. It plays a pivotal role in determining the end behavior of the polynomial. Depending on whether this coefficient is positive or negative, and whether the degree of the polynomial is odd or even, the function will exhibit different behavior as x goes to infinity or negative infinity.

In the context of the given exercise, the leading coefficient is 3, which is attached to the highest degree term x^{11}. This positive leading coefficient tells us that as x becomes very large or very small, the sign of the function’s value will follow the sign of the degree when the degree is an odd number. Since in this case we are investigating the limit as x approaches negative infinity, the positive leading coefficient indicates the function will decrease towards negative infinity.

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of x that appears in the expression with a non-zero coefficient. It provides vital information about the function's shape and its potential number of roots, or solutions. Polynomials with higher degrees can have more complex curves and additional turning points.

When considering limits at infinity, the degree indicates how the value of the function will behave as x grows larger in absolute value. For instance, if the degree is odd, as in the exercise with the polynomial 3x^{11}, we can expect that the sign of the function’s value will ultimately mirror the sign of x. Consequently, an odd-degree polynomial will tend to negative infinity as x approaches negative infinity if its leading coefficient is positive, exactly as concluded in the given exercise.