Problem 21 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{n \rightarrow \infty} 5 x$$

Step-by-Step Solution

Answer

Answer: The limit of the function \(5x\) as \(x\) approaches infinity does not exist because it tends to infinity. $$\lim_{x \rightarrow \infty} 5x = \infty$$

Step 1: Identify the function

The function given is \(f(x) = 5x\). We need to find the limit of this function as \(x\) approaches infinity.

Step 2: Determine the limit

As \(x\) approaches infinity, the value of \(5x\) also approaches infinity, since multiplying an infinitely large number by a positive constant will result in an infinitely large number. Therefore, the limit of the function \(5x\) as \(x\) approaches infinity does not exist because it tends to infinity: $$\lim_{x \rightarrow \infty} 5x = \infty$$

Key Concepts

Infinity in Calculus

In calculus, infinity is a concept rather than a number, and it plays a key role in understanding the behavior of functions. When we talk about a limit approaching infinity, we mean the value of the function increases without bound as the input becomes larger and larger. Consider the function in the original exercise, \( f(x) = 5x \). As \( x \) becomes larger, \( 5x \) also becomes larger without limit. This illustrates how infinity is used in calculus to describe growth that does not settle at any real number.Limits involving infinity may not produce finite real numbers as answers, but they help us understand how functions behave at extreme values. It's essential to recognize when a function's limit approaches infinity, as this determines the direction and growth pattern of the function.

Behavior of Functions

The behavior of functions is fundamental in calculus, revealing how they grow, shrink, or stabilize. Analyzing function behavior involves studying limits, which can illustrate trends as inputs become very large or very small. For the example \(f(x) = 5x\), the behavior is straightforward: as \(x\) increases towards infinity, \(5x\) grows unboundedly. Here are some key points about function behavior:

• Functions can approach infinity, a real number, or even oscillate.
• Linear functions like \(5x\) have a constant rate of increase when multiplied by a positive constant.
• The slope of a function plays a major role in its increase or decrease.

Understanding function behavior helps predict outcomes and identify patterns, especially in real-world scenarios.

Real Numbers

Real numbers are the backbone of calculus, encompassing all rational and irrational numbers we use to measure and describe quantities. They include integers, fractions, and decimals, serving as the domain and range for most functions in calculus.In the context of limits, real numbers are pivotal. They help define endpoints, intermediate values, and the output of most functions except when a function's limit approaches \(\infty\). When examining \(\lim_{x \rightarrow \infty} 5x = \infty\), no real number is the limit; the value becomes infinitely large.Recognizing whether outcomes are real numbers or tend towards infinity is crucial in distinguishing between converging functions, which approach a finite limit, and diverging functions, which do not. Understanding real numbers' role ensures clarity in analyzing mathematical problems.