Determine the following limits at infinity.

lim x→−∞ x^−11

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Identify the function: The function given is \(f(x) = x^{-11}\).

Understand the behavior of the function: As \(x\) approaches \(-\infty\), the base \(x\) becomes a very large negative number.

Consider the exponent: The exponent is \(-11\), which is a negative odd integer.

Apply the rule for negative exponents: \(x^{-11} = \frac{1}{x^{11}}\). This means the function is the reciprocal of \(x^{11}\).

Analyze the limit: As \(x\) approaches \(-\infty\), \(x^{11}\) becomes a very large positive number, so \(\frac{1}{x^{11}}\) approaches \(0\).

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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 in extreme cases, allowing us to determine whether they approach a specific value, diverge, or oscillate. Evaluating limits at infinity is crucial for analyzing horizontal asymptotes and the end behavior of functions.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. The degree of the polynomial, determined by the highest power of the variable, significantly influences its behavior as x approaches infinity or negative infinity. Understanding the properties of polynomial functions is essential for evaluating their limits.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For example, x^−n is equivalent to 1/x^n. This concept is important when analyzing limits involving negative exponents, as it can transform the expression into a more manageable form, particularly when considering limits at infinity.

Determine the following limits at infinity.lim x→−∞ x^−11