91Ó°ÊÓ

Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a particular value. In the original exercise, we explore the limits of the function \( f(x) = \frac{e^2}{1-x} \) as \( x \to +\infty \) and \( x \to -\infty \).

- As \( x \to +\infty \), the denominator \( 1-x \) becomes significantly negative, thus making the fraction approach zero. This is because dividing a constant by a very large negative number results in a value very close to zero.- Similarly, as \( x \to -\infty \), the denominator \( 1-x \) nears positive infinity. Once more, dividing \( e^2 \) by an infinite value approaches zero.

In simpler terms, limits allow us to understand the end behavior of functions and how they behave as inputs extend toward infinity or negative infinity. They're crucial in determining the asymptotic nature of functions and are fundamental in calculus for establishing continuity and defining derivatives.

Asymptotes give important information about how a function behaves as the input values become very large or very small.

- A **vertical asymptote** occurs when the function approaches infinity as the input gets close to certain values. In the function \( f(x) = \frac{e^2}{1-x} \), there is a vertical asymptote at \( x = 1 \). This is because when \( x \) is 1, the denominator becomes zero, making the function value undefined.- On the other hand, **horizontal asymptotes** describe the value that a function approaches as \( x \) goes to infinity or negative infinity. For our function, as discussed earlier, \( f(x) \) approaches zero as \( x \to \pm \infty \), hence the x-axis (\( y = 0 \)) can be considered a horizontal asymptote.

Understanding asymptotes helps with graphing the function and predicting its long-term behavior. They show where the function stabilizes, and where it explodes, providing valuable insights into the function's growth and decay patterns.

Graphing functions provides a visual representation of the mathematical relationships described by equations. In the given function \( f(x) = \frac{e^2}{1-x} \), we perceive its behavior based on its limits and asymptotes.

- **Plotting the Function:** Start by determining the key features—notice where it gets close to an axis or undefined. In this function, it approaches zero as \( x \rightarrow \pm \infty \), verifying our previous limit analysis.- **Identifying Asymptotes:** Include the vertical asymptote at \( x = 1 \) on your graph. This line represents where the function values shoot infinitely high or low, as \( x \) nears 1.- **No Crossing Points:** Interestingly, this function does not intercept the x-axis or y-axis, emphasizing its behavior below zero along the entire curve.

Sketching functions impresses upon us the importance of recognizing asymptotic paths and limit behavior. With these observations, predicting function behavior becomes intuitive, providing a thorough understanding of both local and global function characteristics.