Understanding Limits | Exploring the Behavior of Functions at Infinity and Zero

Two Special Limits

In mathematics, limits are used to describe the behavior of a function or sequence as the input value approaches a particular value

In mathematics, limits are used to describe the behavior of a function or sequence as the input value approaches a particular value. There are two special limits that are frequently encountered: the limit at infinity and the limit at zero.

1. Limit at Infinity:
The limit at infinity represents the behavior of a function or sequence as the input value becomes extremely large. We denote this limit using the symbol ∞. Formally, we say that the limit of a function f(x) as x approaches infinity is denoted as:

lim f(x) = L
x → ∞

This means that as x gets larger and larger, the function f(x) approaches a particular value L. If the limit does not exist, we say the function or sequence diverges.

For example, let’s consider the function f(x) = 1/x. As x approaches infinity, the value of 1/x gets closer and closer to zero. Thus, the limit of f(x) as x goes to infinity is:

lim (1/x) = 0
x → ∞

2. Limit at Zero:
The limit at zero describes the behavior of a function or sequence as the input value approaches zero. We denote this limit using the symbol 0. Formally, we say that the limit of a function f(x) as x approaches zero is denoted as:

lim f(x) = L
x → 0

This means that as x gets closer and closer to zero, the function f(x) approaches a particular value L. Again, if the limit does not exist, we say the function or sequence diverges.

For example, let’s consider the function g(x) = sin(x)/x. As x approaches zero, the value of sin(x)/x approaches 1. Thus, the limit of g(x) as x goes to zero is:

lim (sin(x)/x) = 1
x → 0

Limits at infinity and zero are important concepts in calculus and analysis. They help us analyze the behavior of functions and sequences in extreme cases and have applications in various areas of mathematics, science, and engineering.

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