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

Guida verificata passo dopo passo

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.

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.

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.

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.

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.

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