Determining the Limit of a Function as x Approaches Infinity: A Step-by-Step Guide

limit for f(x) as approaches infinity

To determine the limit of a function as it approaches infinity, we need to evaluate the behavior of the function as the input values become larger and larger

To determine the limit of a function as it approaches infinity, we need to evaluate the behavior of the function as the input values become larger and larger. There are three possible scenarios for the behavior of a function as x approaches infinity:

1. If the function approaches a specific finite value, then the limit is that value. This means that as x gets larger, the function gets closer and closer to a particular number.

2. If the function grows without bound as x increases, then the limit is positive or negative infinity, depending on the direction the function is growing.

3. If the function oscillates between different values as x increases, or if it does not exhibit a clear trend, then the limit does not exist.

To find the limit of a function as x approaches infinity, you can follow these steps:

Step 1: Analyze the highest-power term in the function. If the power is 0, the function is a constant and the limit is that constant as x approaches infinity.

Step 2: If the highest-power term has a positive coefficient, then the function grows without bound as x increases, and the limit is positive or negative infinity depending on the sign of the coefficient.

Step 3: If the highest-power term has a negative coefficient, then the function decreases without bound as x increases, and the limit is again positive or negative infinity depending on the sign of the coefficient.

Step 4: If the highest-power term has a coefficient of zero, you will need to examine the next highest-power term to determine the limit. Repeat steps 1-3 for the next highest-power term.

Step 5: If the function contains trigonometric functions, logarithms, exponential functions, or other more complex terms, additional techniques may be needed to determine the limit.

Let’s work through an example:

Find the limit of the function f(x) = 3x^2 + 5x – 2 as x approaches infinity.

In this case, the highest-power term is x^2, and it has a positive coefficient. Therefore, the function grows without bound as x increases. This means that the limit as x approaches infinity is positive infinity.

So, the limit of f(x) as x approaches infinity is positive infinity.

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