Problem 57 - Calculus of a Single Variable

Question

Using Symmetry to Find Limits If \(f\) is a continuous function such that \(\lim _{x \rightarrow \infty} f(x)=5\) , find, if possible, \(\lim _{x \rightarrow-\infty} f(x)\) for each specified condition. (a) The graph of \(f\) is symmetric with respect to the \(y\) -axis. (b) The graph of \(f\) is symmetric with respect to the origin.

Step-by-Step Solution

Answer

For a function symmetric with respect to the y-axis, \(\lim_{x \rightarrow -\infty} f(x) = 5\). For a function symmetric with respect to the origin, \(\lim_{x \rightarrow -\infty} f(x) = -5\).

Step 1: Determine the limit for the symmetry with respect to the y-axis

To find the limit of the function \(f(x)\) when \(x\) approaches \(-\infty\), use the fact that the function is symmetric with respect to the y-axis. This means that \(f(x) = f(-x)\) for all \(x\). Therefore, if \(\lim_{x \rightarrow \infty} f(x)=5\), then by symmetry, \(\lim_{x \rightarrow -\infty} f(x)\) also equals 5.

Step 2: Determine the limit for the symmetry with respect to the origin

For a function with symmetry with respect to the origin, use the property that \(f(x) = -f(-x)\) for all \(x\). Given \(\lim_{x \rightarrow \infty} f(x)=5\), replace \(x\) with \(-x\) which gives \(-f(x) = 5\). Therefore, the value for the limit as \(x\) approaches \(-\infty\), \(\lim_{x \rightarrow -\infty} f(x)\), will be \(-5\).

Key Concepts

Continuous Functions

A continuous function is a type of function where its graph is an unbroken curve. This means you can draw it without lifting your pen off the paper. Mathematically, a function \( f(x) \) is continuous if for any given point \( c \) in its domain, the limit as \( x \) approaches \( c \) equals the function's value at \( c \). This can be depicted as: \[ \lim_{x \to c} f(x) = f(c) \].
This property is crucial because it guarantees smooth and predictable behavior, which is essential when you're trying to find limits. Continuous functions do not have gaps or jumps, making them easier to analyze when approaching infinity or when considering symmetry.

Symmetry with Respect to the y-axis

Symmetry with respect to the y-axis implies that a function's graph remains unchanged when reflected over the y-axis. Mathematically, a function is y-axis symmetric if \( f(x) = f(-x) \). This means you get the same function value whether you plug in a number or its negative counterpart.

• For example, the function \( f(x) = x^2 \) is symmetric with respect to the y-axis because \( f(-x) = (-x)^2 = x^2 \).
• This property is useful when evaluating limits because it allows you to determine the behavior of a function as \( x \) approaches \( -\infty \) if you already know the behavior as \( x \) approaches \( \infty \).

Symmetry with Respect to the Origin

When a function's graph is symmetric with respect to the origin, it means that if you rotate the function 180 degrees around the origin, the graph looks exactly the same. Mathematically, this is represented by the equation \( f(x) = -f(-x) \).

• An example of such a function is \( f(x) = x^3 \), where \( f(-x) = -(-x)^3 = -x^3 \).
• With this symmetry, if you know that the limit as \( x \) approaches \( \infty \) is a certain value, the limit as \( x \) approaches \( -\infty \) will be its negative.

This kind of symmetry helps in understanding and determining the limits from both ends of the number line effectively.

Behavior of Functions at Infinity

The behavior of functions at infinity refers to how a function behaves as the input values grow larger and larger (either positively or negatively). Typically, this involves finding the limit of the function as \( x \) approaches \( \infty \) or \( -\infty \).

• If we say that \( \lim_{x \to \infty} f(x) = L \), it means that as \( x \) becomes very large, the function value \( f(x) \) gets closer and closer to \( L \).
• This behavior is crucial in evaluating real-world phenomena and understanding asymptotic behaviors of graphs.

Analyzing behavior at infinity helps mathematicians and scientists predict long-term trends in numerical data and graph functions, making it a fundamental concept in calculus.