Problem 88 - Calculus

Question

In Problems \(87-88,\) give an example of: A function \(f(x)\) where \(\lim _{x \rightarrow \infty} f(x)=2\) and \(\lim _{x \rightarrow-\infty} f(x)=-2\)

Step-by-Step Solution

Answer

Example: \( f(x) = 2 \cdot \frac{x}{\sqrt{x^2+1}} \).

Step 1: Understanding the Limit Condition

We need a function \( f(x) \) where \( \lim_{x \to \infty} f(x) = 2 \) and \( \lim_{x \to -\infty} f(x) = -2 \). This means as \( x \) goes to positive infinity, \( f(x) \) should get closer to 2, and as \( x \) goes to negative infinity, \( f(x) \) should approach -2.

Step 2: Choosing a Suitable Function

A suitable function often used for such conditions is a hyperbolic tangent function or a rational function with horizontal asymptotes. We could use a form of the arctangent function: \( f(x) = 2 \cdot \text{tanh}(x) \), but adjusting parameters might be needed for precise limits.

Step 3: Constructing a Modified Function

Let's consider \( f(x) = 2 \cdot \frac{x}{\sqrt{x^2+1}} \). As \( x \to \infty \), \( \frac{x}{\sqrt{x^2+1}} \rightarrow 1 \) leading to \( f(x) \rightarrow 2 \). Similarly, as \( x \to -\infty \), \( \frac{x}{\sqrt{x^2+1}} \rightarrow -1 \) so \( f(x) \rightarrow -2 \).

Step 4: Verification of Limits

Let's verify: For \( x \to \infty \), \( \frac{x}{\sqrt{x^2+1}} \approx \frac{x}{x} = 1 \), thus \( f(x) \approx 2 \times 1 = 2 \). For \( x \to -\infty \), similarly, we get \( f(x) \approx 2 \times (-1) = -2 \). Our chosen function satisfies the given conditions.

Key Concepts

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as the input, typically denoted as \( x \), approaches a critical value like infinity or negative infinity. When discussing asymptotic behavior, we are interested in understanding the end behavior of a function. For instance, which value does the function approach, but not necessarily reach, as the input becomes extremely large or extremely small?

In our example, the function \( f(x) = 2 \cdot \frac{x}{\sqrt{x^2+1}} \) is analyzed for its asymptotic behavior as \( x \) goes to infinity and negative infinity. The function approaches horizontal asymptotes. This means it gets closer and closer to a certain value without necessarily equaling it. In this case, as \( x \to \infty \), the function approaches the asymptote at \( y = 2 \), and as \( x \to -\infty \), it approaches \( y = -2 \).

Asymptotic behavior is crucial in understanding the long-term tendencies of mathematical functions and their graphs. It shows us that while functions may have complex expressions, their long-term growth or decay can be predictable and inform various real-world phenomena.

Infinity in Calculus

Infinity in calculus is a concept that captures the idea of quantities growing without bound. When talking about limits involving infinity, we're examining how functions behave as their inputs become very large or very small. In calculus, this is essential for analyzing a function's end-behavior and determining limits.

In the exercise provided, we see the use of symbolic notation like \( \lim_{x \to \infty} f(x) = 2 \) and \( \lim_{x \to -\infty} f(x) = -2 \). Here, the limit signifies where the function \( f(x) \) is approaching as \( x \) moves toward infinity or negative infinity. This helps us evaluate the 'destination' of the function without the need to evaluate it at infinite points, which isn't possible.

Using limits, calculus allows us to succinctly express and understand tendencies of functions against the concept of the infinite, leading to practical insights for fields requiring assessment of trends and behaviors extending beyond the finite range of standard computation.

Function Analysis

Function analysis involves examining a function's properties, how it behaves, and any significant features it might have. This includes looking at limits, asymptotic behavior, continuity, differentiability, and more. Analyzing functions helps in understanding their graphs, predicting future values, and solving practical problems.

For the function \( f(x) = 2 \cdot \frac{x}{\sqrt{x^2+1}} \), as introduced in the step-by-step solution, we're particularly interested in its limit behavior, which tells us about its end behavior and the horizontal asymptotes.

• Start by noting the formula used here, which combines algebraic manipulation with square roots to drive the function's behavior towards the desired limits.
• Each part of the function — the numerator and the expression within the square root — plays a role in balancing the function towards the limits of '2' and '-2'.

When analyzing such a function, the verification involves checking the polynomial degree, simplifying using limits, and visualizing through graphing to confirm your analytical results. This is a testament to how calculus and algebra come together to tackle complex real-world questions through function analysis.