Finding Limits


In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)


f(x) = 2/x − 3

Verified step by step guidance

1. Step 1: Understand the function f(x) = 2/x - 3. This is a rational function where the term 2/x will approach zero as x approaches infinity or negative infinity.
2. Step 2: Consider the limit as x approaches infinity (x → ∞). As x becomes very large, the term 2/x becomes very small, approaching zero. Therefore, the function f(x) approaches -3.
3. Step 3: Consider the limit as x approaches negative infinity (x → -∞). Similarly, as x becomes very large in the negative direction, the term 2/x also approaches zero. Therefore, the function f(x) approaches -3.
4. Step 4: Visualize the behavior of the function using a graphing calculator or computer. The graph will show that as x moves towards positive or negative infinity, the function approaches the horizontal line y = -3.
5. Step 5: Conclude that the limits are: (a) as x → ∞, the limit of f(x) is -3; (b) as x → -∞, the limit of f(x) is also -3.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are crucial for understanding the behavior of functions at specific points, including infinity. In this context, finding the limit as x approaches infinity or negative infinity helps determine the end behavior of the function.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its input grows very large or very small. For rational functions, this often involves identifying horizontal or vertical asymptotes, which indicate the values the function approaches but may never reach. Understanding this concept is essential for analyzing limits at infinity.

Graphical Interpretation

Graphical interpretation involves using graphs to visualize the behavior of functions, particularly as they approach limits. By plotting the function, one can observe trends and asymptotic behavior, making it easier to understand how the function behaves at extreme values of x. This visual approach can complement analytical methods in finding limits.