Problem 56 - Essential Calculus

Question

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 to the \(y\) -axis. (b) The graph of \(f\) is symmetric to the origin.

Step-by-Step Solution

Answer

For symmetric functions across the y-axis, \( \lim_{{x \to - \infty}} f(x) = 5 \). While for symmetric functions across the origin, \( \lim_{{x \to - \infty}} f(x) = -5 \).

Step 1: Calculate limit for symmetry across y-axis

For a function to be symmetric around the y-axis, it means it follows the rule \(f(-x) = f(x)\). In such case, the limit as \(x\) approaches negative infinity should be similar to when \(x\) approaches positive infinity since the two sides mirror each other. Therefore, as \(x\) approaches negative infinity, the limit will also be 5 which becomes \( \lim_{{x \to -\infty}} f(x) = 5 \).

Step 2: Calculate limit for symmetry across the origin

For a function to be symmetric around the origin, it implies that it obeys the rule \(f(-x) = -f(x)\). For this case, as \(x\) approaches positive infinity, we have the limit of \(f(x)\) as 5. Since the function is odd and thus changes sign, if we approach negative infinity, we should change the sign of the limit as well. Therefore, for \(f(-x)\), the limit will become negative 5, or \( \lim_{{x \to -\infty}} f(x) = -5 \).

Key Concepts

Continuous Function

A continuous function is essential in calculus, similar to a smooth road with no bumps or breaks. When we talk about a function being continuous over a certain interval, we mean you can draw it without lifting your pen from the paper. In technical terms, a function \( f(x) \) is continuous at a point \( a \) if three conditions are met:

• The function \( f(x) \) is defined at \( x = a \).
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) is equal to \( f(a) \).

These conditions ensure there are no sudden jumps or holes in the graph of the function. Understanding these properties can make it easier to predict behavior of functions as they move towards infinity or negative infinity.When we are looking at the behavior at infinity and using continuous function properties, we can say that if \( \lim_{x \to \infty} f(x) \) is 5, the graph of the function settles to a value of 5 as \( x \) takes on sufficiently large positive values. This property will hold unless additional conditions, such as symmetries, suggest otherwise.

Symmetry in Calculus

Symmetry plays a critical role in understanding functions, especially when calculating limits. In calculus, recognizing different types of symmetry can simplify complex problems. There's often less confusion if we focus on the two main types here:- **Symmetry with respect to the \( y \)-axis:** - When a function \( f(x) \) is symmetric about the \( y \)-axis, it means \( f(-x) = f(x) \). This situation reflects itself in the limits at infinity. If we know \( \lim_{x \to \infty} f(x) = 5 \), then symmetry implies that \( f(x) \) also approaches 5 as \( x \to -\infty \). - **Symmetry with respect to the origin:** - In this case, \( f(-x) = -f(x) \). It's often called odd symmetry. So, if the limit as \( x \to \infty \) is 5, the corresponding limit as \( x \to -\infty \) would flip sign due to the symmetry, resulting in \( \lim_{x \to -\infty} f(x) = -5 \).Understanding these symmetrical properties helps make predictions on how functions behave over intervals and can be especially helpful when working with complex functions that might be otherwise hard to decode.

Limits at Infinity

The concept of limits at infinity involves understanding what value a function approaches as the input grows larger towards positive or negative infinity. It's like watching a car travel down a road; you notice the pattern it follows as it moves further away. Mathematically, \( \lim_{x \to \infty} f(x) = L \) implies that as \( x \) increases to infinity, \( f(x) \) gets arbitrarily close to \( L \). Similarly, \( \lim_{x \to -\infty} f(x) \) involves understanding what happens as the function moves towards negative infinity. What do these limits tell us?

• If a function approaches a specific value, it suggests a horizontal asymptote at that value.
• The graph of the function "flattens out" as \( x \) becomes very large in either direction.
• Utilizing symmetry can also help clarify these limits, as we noted earlier.

By evaluating limits at infinity, mathematicians and students alike can gain insights into the end behavior of functions, revealing patterns and predictions that are crucial for calculus and beyond.