Problem 58 - Calculus: Graphical, Numerical, Algebraic

Question

Writing to Learn Let \(L\) be a real number, lim \(_{x \rightarrow c} f(x)=L\) and \(\lim _{x \rightarrow c} g(x)=\infty\) or \(-\infty .\) Can \(\lim _{x \rightarrow c}(f(x)+g(x))\) be determined? Explain.

Step-by-Step Solution

Answer

The limit \(\lim _{x \rightarrow c}(f(x)+g(x))\) cannot be determined as a finite real number. It will either be infinity or negative infinity.

Step 1: Understanding the problem

Here, we are given \(\lim _{x \rightarrow c} f(x)=L\) and \(\lim _{x \rightarrow c} g(x)=\infty\) or \(-\infty\). We are asked to determine the sum of these limits, or in other words, \(\lim _{x \rightarrow c}(f(x)+g(x))\) . Also, remember \(L\) is a real number.

Step 2: Determine the sum of the limits

Generally, if the limit of a function as \(x\) approaches a certain value is infinity or negative infinity, this denotes the function's behavior of increasing or decreasing without bound respectively, as \(x\) approaches that certain value. When we add a finite number (in our case \(L\), which is a real number) to infinity or negative infinity, the sum is still infinity or negative infinity respectively. The presence of the finite number does not change the result.

Step 3: Conclusion

So, in our problem, \(\lim _{x \rightarrow c}(f(x)+g(x))\) cannot be determined as a finite real number, because adding a finite real number to infinity or negative infinity still results in infinity or negative infinity respectively.

Key Concepts

Real Numbers

In mathematics, real numbers include all the numbers on the number line. This comprises all the rational numbers, like fractions and integers, and all the irrational numbers, which cannot be expressed as fractions. Real numbers are fundamental in calculus as they describe the output of functions, limits, and other mathematical operations.
* They are typically denoted by symbols like \( L \) in calculus. This \( L \) represents a specific real number, such as the limit of a function as it approaches a particular point.* Real numbers are finite, meaning they have a definite value that can be pinpointed on the number line.

In our exercise, we are dealing with \( L \) as a real number. This plays a crucial role when determining limits, as it sets a finite value against which infinite behavior, like that of \( g(x) \), is contrasted.

Infinity in Calculus

Infinity is a concept in calculus that describes a quantity without bound or end. It represents a value larger than any real number, which can be approached in equations but never actually reached.

• In calculus, we often use infinity to describe the behavior of functions as they grow beyond any finite value, like \( \lim_{x \to c} g(x) = \infty \) or \( -\infty \).
• This means that as \( x \) approaches a certain value, \( g(x) \) either increases or decreases without limit.
• The presence of infinity means we cannot pinpoint a precise real-number value for some operations, such as the addition in our problem.

When we add a real number \( L \) to infinity, the result is still infinity, as the finite real part does not impact the unbounded nature of infinity.
Therefore, when trying to determine \( \lim_{x \to c} (f(x) + g(x)) \), the inclusion of infinity means this sum cannot resolve into a straightforward, finite value.

Functions and Limits

A function in calculus defines a relationship between an input and an output. When we talk about limits, we describe the value that a function approaches as the input nears a certain point.
* Limits are crucial, especially when dealing with functions that may not have defined values at certain points but still approach a particular number.* If a limit, such as \( \lim_{x \to c} f(x) = L \), is a real number, it suggests that as \( x \) approaches \( c \), \( f(x) \) gets closer and closer to \( L \).However, when we have one function with a limit of a real number and another with a limit of infinity, their combined limit becomes complex. Infinity dominates over any finite values, making it impossible to achieve a real-numbered sum.
Hence, understanding how limits work with different types of functions helps us navigate scenarios where infinity plays a significant role, like in our problem.