Using the Formal Definitions


Use the formal definitions of limits as x → ±∞ to establish the limits in Exercises 91 and 92.


If f has the constant value f(x) = k, then lim x → ∞ f(x) = k.

Verified step by step guidance

1. Step 1: Understand the formal definition of a limit as x approaches infinity. The limit of a function f(x) as x approaches infinity is the value that f(x) gets closer to as x becomes very large.
2. Step 2: Recognize that if f(x) is a constant function, meaning f(x) = k for all x, then as x approaches infinity, f(x) remains constant at k.
3. Step 3: Apply the formal definition of limits. For any ε > 0, there exists a number N such that for all x > N, the absolute difference |f(x) - k| is less than ε.
4. Step 4: Since f(x) = k for all x, the absolute difference |f(x) - k| is 0, which is always less than any positive ε. Therefore, the condition of the limit definition is satisfied.
5. Step 5: Conclude that the limit of f(x) as x approaches infinity is k, because the function value does not change and remains equal to k for all x.

Key Concepts

Limit of a Function

The limit of a function describes the behavior of that function as the input approaches a certain value. In the context of limits as x approaches infinity, it examines how the function behaves as x grows larger and larger. Understanding limits is fundamental in calculus, as it lays the groundwork for concepts such as continuity, derivatives, and integrals.

Formal Definition of a Limit

The formal definition of a limit, often expressed using epsilon-delta notation, provides a rigorous way to define what it means for a function to approach a certain value as the input approaches a specific point. For limits at infinity, this definition helps establish that for every small positive number (epsilon), there exists a corresponding value of x (N) such that for all x greater than N, the function's value is within epsilon of the limit.

Constant Function Limits

A constant function is one where the output value remains the same regardless of the input, expressed as f(x) = k. The limit of a constant function as x approaches infinity is simply the constant value itself, lim x → ∞ f(x) = k. This concept is crucial for understanding how functions behave at extreme values and simplifies the analysis of limits in calculus.