Question

A function \(f\) is even if \(f(-x)=f(x)\) for all \(x\) in the domain of \(f\). If \(f\) is even, with \(\lim _{n \rightarrow 2^{+}} f(x)=5\) and \(\lim _{x \rightarrow 2^{-}} f(x)=8,\) find the following limits. a. \(\lim _{x \rightarrow-2^{+}} f(x)\) b. \(\lim _{x \rightarrow-2^{-}} f(x)\)

Short answer

Question: Given that \(f\) is an even function and we are given \(\lim _{x \rightarrow 2^{-}} f(x) =8\) and \(\lim _{x \rightarrow 2^{+}} f(x)=5\), find: a. \(\lim _{x \rightarrow -2^{+}} f(x)\) b. \(\lim _{x \rightarrow -2^{-}} f(x)\) Answer: a. \(\lim _{x \rightarrow -2^{+}} f(x) = 8\) b. \(\lim _{x \rightarrow -2^{-}} f(x) = 5\)

Step-by-step solution

Recall the property of an even function

Given that \(f\) is an even function, we know that \(f(-x) = f(x)\) for all \(x\) in the domain of \(f\).

Find the limit as \(x\) approaches -2 from the right

We know that \(\lim _{x \rightarrow 2^{-}} f(x)=8\), and we're asked to find \(\lim _{x \rightarrow -2^{+}} f(x)\). Since \(f\) is even, \(f(-x) = f(x)\). Therefore, we can replace the -2 with 2 in the limit as follows: \(\lim _{x \rightarrow -2^{+}} f(x) = \lim _{x \rightarrow (-1)(2^{-})} f(x) = \lim _{x \rightarrow 2^{-}} f(-x)\) But given that \(f\) is even, \(f(-x)=f(x)\). Thus: \(\lim _{x \rightarrow -2^{+}} f(x) = \lim _{x \rightarrow 2^{-}} f(x)=8\)

Find the limit as \(x\) approaches -2 from the left

We know that \(\lim _{x \rightarrow 2^{+}} f(x)=5\), and we're asked to find \(\lim _{x \rightarrow -2^{-}} f(x)\). Similar to the previous step, since \(f\) is even, \(f(-x) = f(x)\). Therefore, we can replace the -2 with 2 in the limit as follows: \(\lim _{x \rightarrow -2^{-}} f(x) = \lim _{x \rightarrow (-1)(2^{+})} f(x) = \lim _{x \rightarrow 2^{+}} f(-x)\) Again, since \(f\) is even, \(f(-x)=f(x)\). Thus: \(\lim _{x \rightarrow -2^{-}} f(x) = \lim _{x \rightarrow 2^{+}} f(x)=5\) So, the final answers are: a. \(\lim _{x \rightarrow -2^{+}} f(x) = 8\) b. \(\lim _{x \rightarrow -2^{-}} f(x) = 5\)

Key concepts

Understanding the Limit of a Function

A function's limit helps us understand its behavior as the input approaches a particular value. It provides insight into the function's behavior without actually reaching that point. In simple terms, if we have a function \(f(x)\), and we're interested in its behavior as \(x\) approaches a certain value, we can say \( \lim_{x \to a} f(x) = L \). This means that as \(x\) gets closer to \(a\), \(f(x)\) approaches \(L\).

Let's break it down further. Consider the notation \( \lim_{x \to 2^{+}} f(x) \). The "plus" sign indicates we are approaching from the right of 2. This tells us the function values as \(x\) approaches 2 from values greater than 2. Similarly, \( \lim_{x \to -2^{-}} f(x) \) is about approaching from the left of -2, or values less than -2.

By understanding limits, we can predict the output of functions even in points they are not defined, effectively making this concept fundamental in calculus and real-world applications. For example, when analyzing signals, stability of systems, or behaviors of complex systems.

Exploring the Properties of Functions

Functions have various properties, and understanding these properties helps us work with functions more effectively. One essential property is whether a function is even or odd. A function \( f(x) \) is even if \( f(-x) = f(x) \) for every \( x \) in its domain. This symmetry means the graph of an even function is mirrored across the y-axis.

• Symmetry: Even functions have a symmetric graph around the y-axis.
• Helpful for limits: Knowing a function is even can simplify limit calculations, as seen in the exercise.
• Applications: Many physics problems involve even functions, reflecting symmetric physical laws.

When working with limits, the knowledge that a function \( f \) is even tells us that \( \lim_{x \to a} f(x) = \lim_{x \to -a} f(x) \). This property has been critical in finding limits in the given exercise because it allowed us to use known limits of positive \( x \) to find limits of negative \( x \). Utilizing function properties can simplify complex problems and yield solutions efficiently.

Understanding Calculus and Its Importance

Calculus is a powerful branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It provides the tools needed to understand and describe change and motion in mathematical terms. By exploring calculus, we gain the ability to:

• Analyze functions: Understanding limits and properties helps predict function behavior.
• Model real-life phenomena: From physics to economics, calculus lets us model change systems.
• Optimize systems: Calculus techniques can help find maxima and minima, optimizing performance.

A fundamental aspect of calculus is understanding how functions behave through limits. This involves knowing how they approach or exceed certain values, giving us a detailed picture of their continuity and smoothness. The knowledge of calculus empowers us to solve complex problems in various scientific fields and everyday life situations. Calculus expands our analytical capabilities, making it a vital tool for anyone dealing with continuous systems or needing to predict future behavior based on current data.