Question

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

Short answer

Answer: The limit of the function \(x^{-11}\) as \(x\) approaches negative infinity is 0.

Step-by-step solution

Analyze the function

First, let's rewrite the given function so that it's easier to understand its behavior as \(x\) goes to negative infinity. Notice that \(x^{-11} = \frac{1}{x^{11}}\). As \(x\) goes to negative infinity, the denominator, \(x^{11}\), is becoming a very large negative number. However, since 11 is an odd number, the negative sign remains in the denominator, making the overall fraction having an alternating sign. Now let's find the limit of the function:

Evaluate the limit

To find the limit of the function as x approaches negative infinity, we can simply analyze the behavior of the function based on the observation we made in Step 1. $$\lim_{x \rightarrow -\infty} x^{-11} = \lim_{x \rightarrow -\infty} \frac{1}{x^{11}}$$ As x goes to negative infinity, the denominator \(x^{11}\) becomes a very large negative number, while the numerator remains constant at 1. Therefore, the function \(\frac{1}{x^{11}}\) approaches zero because the denominator increases indefinitely, and the sign of the fraction will alternate between positive and negative values. This means the limit of the function as x approaches negative infinity is 0: $$\lim_{x \rightarrow -\infty} \frac{1}{x^{11}} = 0$$ So the limit of the function \(x^{-11}\) as \(x\) approaches negative infinity is 0.

Key concepts

Negative Infinity

When we talk about negative infinity in mathematics, we refer to a concept rather than a specific number. It represents a number that is smaller than any other real number. It is used mainly when discussing limits or behaviors of functions as values drop endlessly below zero.

In the context of limits, approaching negative infinity means observing what happens to the function as the value of the variable heads towards negative infinity. Consider the behavior of traditional graphing. As we move left along the x-axis, the values may either rise, fall, or level out, giving a clue about the function's end behavior.

Understanding negative infinity can aid in determining the behavior of functions, especially rational functions, as their x-values become increasingly negative.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They are an essential type of function in calculus and are defined as a ratio \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. The domain excludes values where the denominator, \(Q(x)\), is zero since division by zero is undefined.

In the case of limits at infinity, like \(x^{-11}\), we often rewrite expressions in terms of powers. Here, \(x^{-11}\) turns into \(\frac{1}{x^{11}}\), making it a particular case of a rational function that simplifies understanding by highlighting how the function behaves for large values of \(x\).

Analyzing the powers involved in these rational functions can tell us if the function grows, shrinks, or tilts in a certain direction as \(x\) approaches infinity or negative infinity. A keen grasp on how rational functions react to extreme values allows a determination of limits.

Behavior of Functions as x Approaches Infinity

Behavior at infinity is a central topic within calculus when evaluating limits. It's all about analyzing how functions behave as \(x\) moves to either positive or negative infinity. This behavior helps in predicting the trend of the function and knowing if it settles, diverges, or oscillates.

For instance, in rational functions, the degree of the numerator versus the denominator dictates its end behavior. If the degree of the denominator exceeds that of the numerator, as \(x\) approaches infinity or negative infinity, the function approaches zero. This is because the denominator increases much faster, diminishing the fraction.

In the example of \(x^{-11} = \frac{1}{x^{11}}\), as \(x\) tends to negative infinity, the function heads to zero. Each increase in the negative value of \(x\) results in \(x^{11}\) becoming a gigantically large negative. This causes the entire fraction to become minute, fluctuating slightly in sign, yet zero remains the logical outcome as the limit. Understanding these dynamics enables better problem-solving and comprehension of advanced mathematical concepts.