91Ó°ÊÓ

In calculus, limits help us understand the behavior of functions as they approach specific points. They are crucial for defining continuity, derivatives, and integrals. Consider the function \( f(x) = \frac{1}{1 + e^{1/x}} \). It demonstrates different behaviors as \(x\) approaches 0 from the right (* \(x \to 0^+\)) and from the left (* \(x \to 0^-\)).

• For \(x \to 0^+\): As \( \frac{1}{x} \to +\infty\), \( e^{1/x} \to +\infty\), turning \(f(x)\) into \(\frac{1}{1 + \infty} = 0\).
• For \(x \to 0^-\): As \( \frac{1}{x} \to -\infty\), \( e^{1/x} \to 0\), making \(f(x)\) approach \(\frac{1}{1 + 0} = 1\).

Assessing limits as \(x\) heads towards infinity, either positive or negative, also provides insight. Here, \(e^{1/x} \to 1\) and \(f(x) \to \frac{1}{2}\). Observing how \(f(x)\) behaves in these extremes sets the stage to explore its broader properties.

Continuity occurs when a function has no interruptions or breaks at any point within its domain. In our function \(f(x)\), continuity is disrupted at \(x = 0\) due to the undefined term \(1/x\).

• A function is continuous at a point if the limit of the function as it approaches the point from both directions equals the function's value at that point.
• For \(f(x)\) at \( x = 0\), the right-hand limit is 0 and the left-hand limit is 1. Since these limits differ, \(f(x)\) is not continuous at \( x = 0 \).

Elsewhere, \(f(x)\) is continuous because it behaves predictably without abrupt changes as noted from its reactions approaching infinity. Continuous functions allow us to use calculus to predict and analyze their behavior better.

The derivative of a function represents the rate at which the function's value changes as its input changes, essentially depicting the function's slope at any given point. Calculating the derivative \(f'(x)\) for \(f(x) = \frac{1}{1 + e^{1/x}}\) requires applying both the chain rule and quotient rule.

Calculating the Derivative

Set \(u = 1 + e^{1/x}\). By applying the rules:

• The derivative of \(f(x)\) is \( f'(x) = -\frac{1}{u^2} \times u'(x)\).
• For \(u'(x)\), we use \( u'(x) = -\frac{1}{x^2} e^{1/x} \).
• Thus, \( f'(x) = \frac{e^{1/x}}{(1 + e^{1/x})^2 x^2} \).

The derivative's behavior at \(x = 0\) is erratic due to the expression exploding, meaning limits for \(f'(x)\) as \(x\) approaches 0 don't exist. Understanding derivatives is essential for sketching the graph of \(f(x)\) and identifying points of rapid change.

Graphing functions reveals their overall structure and behavior, illustrating important characteristics such as limits, continuity, and derivatives. For \(f(x) = \frac{1}{1 + e^{1/x}}\), drawing its graph helps visualize how it behaves near key points:

• As \(x \to 0^+\), \(f(x)\) approaches 0, indicated by the graph nearing the x-axis from the right.
• Conversely, approaching from left (* \(x \to 0^-\)) brings \(f(x)\) to 1, making the graph meet the line \(y = 1\).
• As \(x\) extends towards infinity or negative infinity, \(f(x)\) stabilizes at 0.5, seen as an asymptote parallel to \(y = \frac{1}{2}\).

Thus, graphing involves capturing these traits to accurately present \(f(x)\)'s behavior. Observing the derivative \(f'(x)\) alongside can further show the rate of change, highlighting steeper graph sections where shifts happen more rapidly.