Post 2: Understanding the Definition of Limits

In calculus, the concept of limits is fundamental in understanding the behavior of functions and their values as they approach certain points or infinity. By understanding limits, we can analyze the behavior of functions, solve calculus problems, and make predictions in various real-world scenarios.

Definition of Limits

The limit of a function, as x approaches a specific value, represents the value that the function approaches as it gets arbitrarily close to that particular point. We denote the limit of a function f(x) as x approaches a as follows:

lim(x -> a) f(x) = L

Here, L represents the limit of the function f(x) as x approaches a.

One-Sided and Two-Sided Limits

We can distinguish between one-sided limits and two-sided limits:

1. One-Sided Limits: We consider one-sided limits when x approaches a from only one direction (either left or right). The left-hand limit is denoted as:

lim(x -> a-) f(x)

This represents the limit of the function f(x) as x approaches a from the left side. Similarly, the right-hand limit is denoted as:

lim(x -> a+) f(x)

This represents the limit of the function f(x) as x approaches a from the right side.

2. Two-Sided Limits: We use two-sided limits when x approaches a from both directions (left and right). The two-sided limit is denoted as:

lim(x -> a) f(x)

This limit represents the overall behavior of the function f(x) as x approaches a, considering both left and right approaches.

Examples

Let's explore a few examples to better understand the concept of limits:

Example 1: Find the limit of the function f(x) = 2x - 1 as x approaches 3.

We need to find lim(x -> 3) (2x - 1)

By direct substitution:
lim(x -> 3) (2x - 1) = 2 * 3 - 1 = 5

Therefore, lim(x -> 3) (2x - 1) = 5

Example 2: Find the limit of the function g(x) = (x^2 - 1) / (x - 1) as x approaches 1.

We need to find lim(x -> 1) [(x^2 - 1) / (x - 1)]

By factoring:
lim(x -> 1) [(x^2 - 1) / (x - 1)] = lim(x -> 1) [(x - 1)(x + 1) / (x - 1)]

Now, cancel out the common factor (x - 1) in the numerator and denominator:
lim(x -> 1) [(x - 1)(x + 1) / (x - 1)] = lim(x -> 1) (x + 1)

By direct substitution:
lim(x -> 1) (x + 1) = 1 + 1 = 2

Therefore, lim(x -> 1) [(x^2 - 1) / (x - 1)] = 2

These examples demonstrate the process of evaluating limits. By understanding the definition of limits and utilizing appropriate techniques, we can accurately determine the behavior of functions as they approach specific values. This knowledge is crucial in various calculus applications, including finding derivatives, determining rates of change, and analyzing the behavior of functions in real-world contexts.