Problem 58 - Calculus: Early Transcendentals

Question

Transcendental functions Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=2^{x}$$

Step-by-Step Solution

Answer

Answer: The end behavior of the function $$f(x) = 2^x$$ is as follows: - As x approaches positive infinity, the function approaches infinity. - As x approaches negative infinity, the function approaches 0. There is a horizontal asymptote at $$y = 0$$.

Step 1: Analyze Limit as x approaches positive infinity

To find the behavior of the function as x approaches positive infinity, we evaluate the limit as x approaches positive infinity: $$\lim_{x \to \infty} 2^x$$ Since 2 is greater than 1, $$2^x$$ will grow without bound as x increases. Therefore, the limit as x approaches positive infinity is: $$\lim_{x \to \infty} 2^x = \infty$$

Step 2: Analyze Limit as x approaches negative infinity

Now we will analyze the limit as x approaches negative infinity: $$\lim_{x \to -\infty} 2^x$$ Since 2 is greater than 1, $$2^x$$ will approach 0 as x approaches negative infinity. Therefore, the limit as x approaches negative infinity is: $$\lim_{x \to -\infty} 2^x = 0$$

Step 3: Identify Asymptotes

As x approaches negative infinity, the function approaches 0. This indicates a horizontal asymptote at $$y = 0$$.

Step 4: Sketch the graph

To sketch the graph of $$f(x) = 2^x$$, keep in mind the behavior identified in steps 1-3: 1. The graph increases without bound as x approaches positive infinity. 2. The graph approaches 0 as x approaches negative infinity (horizontal asymptote at $$y = 0$$). The graph starts close to the x-axis on the left and increases as x goes to the right, while always remaining above the x-axis due to the asymptote. In conclusion, the end behavior of the given transcendental function is as follows: - As x approaches positive infinity, the function approaches infinity. - As x approaches negative infinity, the function approaches 0, with a horizontal asymptote at $$y = 0$$.

Key Concepts

End Behavior Analysis

Understanding the end behavior of functions is crucial in calculus. It tells us what happens to the function's values as the independent variable either increases without bound or decreases without bound. For the transcendental function f(x) = 2^x, we analyzed the end behavior by taking limits at positive and negative infinity.

The function 2^x increases exponentially as x increases. When approaching positive infinity, since two is a base greater than one, the function's values also head towards infinity. This is symbolized by the mathematical expression \( \lim_{{x \to \infty}} 2^x = \infty \). Conversely, when x approaches negative infinity, the function 2^x gets closer and closer to zero but never touches it, creating a horizontal asymptote at y = 0 expressed as \( \lim_{{x \to -\infty}} 2^x = 0 \).

Visualizing End Behavior

On a graph, this end behavior indicates that the curve will rise sharply to the right (as x goes to positive infinity) and gently flatten out as it moves left (as x goes to negative infinity), never crossing the horizontal asymptote.

Limits in Calculus

Limits form the fundamental cornerstone of calculus, helping us to describe the behavior of functions as they approach specific points or infinity. When we say \( \lim_{{x \to c}} f(x) = L \), we mean that the function f(x) gets closer and closer to a value L as x approaches c.

In our exercise with the function f(x) = 2^x, we've used limits to understand how the function behaves at the extremes. The concept of limits extends beyond just finding the value that a function approaches; it also helps us to determine continuity, define derivatives, and solve problems involving indeterminate forms such as 0/0 or \infty/\infty.

Applying Limits

For transcendental functions, which aren't algebraic, limits help us to grasp their often complex behavior. We apply the rules and properties of limits to evaluate the end behavior of functions, including those involving exponential growth or decay, just as we did with 2^x.

Graphing Exponential Functions

Exponential functions, like f(x) = 2^x, are an important class of transcendental functions characterized by their variable exponents. Graphing these functions provides a visual interpretation of the function's characteristics, such as growth rate, end behavior, and any asymptotes.

When graphing 2^x, we start by plotting a few key points to understand its shape. Exponential functions always pass through the point (0, 1), since any non-zero base raised to the power of zero equals one. As we move to the right along the x-axis (increasing x), the y-values increase sharply, reflecting exponential growth. Moving left (decreasing x), the graph gets closer to the x-axis but never touches it due to the horizontal asymptote at y = 0.

Sketching for Clarity

A sketch should accurately depict these characteristics and include asymptotes. Remember that the function f(x) = 2^x will always have a horizontal asymptote along the x-axis for negative values of x. A simple sketch helps to visualize complex concepts like end behavior and limits, reinforcing the analytical results obtained from the calculations.