Limits

Let us begin with an example. This is the graph of f(x) = x² + 3x - 1. At x = 1, the graph has a y-value of 3. So, we say that the limit of the function at x = 1 is 3.

If you want to calculate the limit algebraically, substitute x = 1 in the function.

f(x) = x² + 3x - 1

f(1) = 1² + 3*1 - 1

f(1) = 3

So, the value of the function at x = 1 is called its limit at x = 1. This is written as

lim

x → 1

x² + 3x - 1 = 3

Not all functions have a specific y value for a specific x-value, as the above one. In such functions, the limit is taken as the y-value around towards which the graph is approaching from both sides of the x value. For example, the graph of the following function does not exist at x = 1. But, looking at its graph, you can tell that it is approaching y = 10 when you move in nearer and nearer to x = 1. Thus, although the value of the function does not exist at x = 1, its limit is said to be equal to 10 as x approaches 1.

Difference between value of a function and limit of a function is that the limit is the value which the function is approaching at a particular value of x, while the value of a function is its exact value at a particular value of x.