Question

Determine the following limits at infinity. $$\lim _{x \rightarrow-\infty} 2 x^{-8}$$

Short answer

Answer: The limit of the function \(2x^{-8}\) as \(x\) approaches negative infinity is 0.

Step-by-step solution

Write down the given function

We have the function, $$f(x) = 2x^{-8}$$ Our goal is to find the limit of this function as \(x\) approaches negative infinity: $$\lim_{x \rightarrow -\infty} 2x^{-8}$$

Rewrite the function with a positive exponent#

Rewrite the function with a positive exponent by using the property that \(x^{-n} = \frac{1}{x^n}\): $$f(x) = 2x^{-8} = \frac{2}{x^8}$$ Now, the limit can be written as: $$\lim_{x \rightarrow -\infty} \frac{2}{x^8}$$

Evaluate the limit as x approaches negative infinity#

As \(x\) becomes infinitely large in the negative direction (\(x \rightarrow -\infty\)), the denominator \(x^8\) becomes infinitely large as well. Since the exponent 8 is even, even if \(x\) is negative, raising it to the power of 8 results in a positive value. So, we have: $$\lim_{x \rightarrow -\infty} \frac{2}{x^8} = \frac{2}{\infty}$$

Determine the final limit value#

When dividing a finite value (in this case, 2) by an infinitely large value (in this case, the denominator \(x^8\) as \(x\) approaches negative infinity), the quotient becomes infinitely small, approaching 0. Thus, the final limit value is: $$\lim_{x \rightarrow -\infty} 2x^{-8} = \lim_{x \rightarrow -\infty} \frac{2}{x^8} = 0$$

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Its two primary divisions, differential calculus and integral calculus, are linked by the fundamental theorem of calculus.

Understanding limits, like those at infinity, is essential in calculus because they form the foundation for defining derivatives and integrals. Calculus enables us to investigate how functions behave as they are influenced by infinitesimally small changes, and how they accumulate quantities over intervals. The calculations of areas, volumes, and rates of change all rely on the core principles of calculus.

Limit Evaluation

Limit evaluation is a foundational concept in calculus that deals with the behavior of a function as its input approaches a particular value or infinity. Practically, it helps to understand function behavior near points that may not be clearly defined, like discontinuities or towards infinity.

In the given exercise, the process of evaluating a limit starts with understanding the function form and then simplifying or transforming that function as needed. Finally, considering the behavior of the terms as the variable approaches the concerned point, like \(x \rightarrow -\infty\), contributes to finding the limit's value.

Negative Infinity

The concept of negative infinity, often denoted as \( -\infty \), comes into play when discussing the limits of functions as the input values decrease without bound. Unlike a positive infinity where values grow larger, negative infinity pertains to values that fall increasingly lower.

It's essential to understand that infinity is a concept, not a number, and expressions like \(x \rightarrow -\infty\) are talking about a trend rather than a destination. In limit evaluation, approaching negative infinity reflects a notion where the input variable within a function becomes indefinitely small in the negative direction.

Exponent Properties

Exponent properties, also known as the laws of exponents, are rules that govern the operations on numbers with exponents. They are fundamental in simplifying expressions involving powers and are crucial when evaluating limits, particularly at infinity.

In the context of limits at infinity, the negative exponent in \(2x^{-8}\) can be transformed using the property that \(x^{-n} = \frac{1}{x^n}\). Understanding how exponents work plays a vital role in determining the behavior of a function as its variable heads towards negative or positive infinity. By mastering exponent properties, simplifying functions to evaluate limits becomes more intuitive.