Understanding Left-Hand Limits in Calculus | Explained with Examples

One sided limit coming from the left

In calculus, a one-sided limit is a limiting process that considers the behavior of a function as it approaches a specific value from either the left side or the right side

In calculus, a one-sided limit is a limiting process that considers the behavior of a function as it approaches a specific value from either the left side or the right side. The one-sided limit from the left, also known as the left-hand limit, is denoted as:

lim (x→c-) f(x)

Here, c is a specific value, and the “x→c-” indicates that we are approaching c from the left side. This means that we are considering the behavior of the function as x values approach c from values that are smaller than c.

To understand the left-hand limit, let’s consider an example. Suppose we have the function f(x) = x^2. If we want to find the one-sided limit from the left at x = 2, we need to evaluate the following:

lim (x→2-) x^2

To do this, we substitute values slightly smaller than 2 into the function and observe the behavior of the function as these values get closer to 2. For example, we may choose 1.5, 1.9, 1.99, and so on, to see how f(x) behaves as x approaches 2 from the left.

If the function gets arbitrarily close to a specific value as x approaches 2 from the left, then that value is the left-hand limit at x = 2. In the case of f(x) = x^2, as x approaches 2 from the left (smaller values), f(x) approaches 4. Therefore, the left-hand limit at x = 2 is 4:

lim (x→2-) x^2 = 4

It is important to note that the left-hand limit is not concerned with the actual value of the function at x = c, but rather its behavior as x approaches c from the left side.

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