Understanding Limit as X Approaches Infinity and Zero in Mathematics

Two Special Limits

In mathematics, limits are an important concept that allow us to understand the behavior of functions as they approach a certain point

In mathematics, limits are an important concept that allow us to understand the behavior of functions as they approach a certain point. There are various types of limits, but in this answer, we will focus on two special types: limit as x approaches infinity and limit as x approaches zero.

1. Limit as x approaches infinity:
The limit as x approaches infinity, denoted as lim(x → ∞), examines the behavior of a function as the independent variable approaches positive infinity. To evaluate this limit, we observe the values of the function as x becomes larger and larger.

Let’s suppose we have a function f(x) = 3x^2 – 2x + 1. To find the limit as x approaches infinity for this function, we consider the leading term, which in this case is 3x^2. As x becomes larger, the influence of the other terms (-2x + 1) becomes negligible compared to 3x^2.

Therefore, lim(x → ∞) 3x^2 – 2x + 1 = lim(x → ∞) 3x^2 = ∞. This means that as x approaches infinity, the value of the function also goes to infinity.

2. Limit as x approaches zero:
The limit as x approaches zero, denoted as lim(x → 0), analyzes the behavior of a function as the independent variable approaches zero on the number line. This limit helps us understand the function’s behavior near zero.

For example, let’s consider the function g(x) = (2x + 1)/(x – 3). To find the limit as x approaches zero for this function, we substitute x = 0 into the function and evaluate:

lim(x → 0) (2x + 1)/(x – 3) = (2(0) + 1)/(0 – 3) = 1/(-3) = -1/3.

Therefore, lim(x → 0) (2x + 1)/(x – 3) = -1/3. This indicates that as x approaches zero, the value of the function approaches -1/3.

By understanding and computing these special types of limits, we gain insight into the behavior of functions at extreme points, such as infinity or zero.

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