This time, notice that the value of the function gets smaller and smaller, as gets closer and closer to 0 with smaller values i.e., from the left side, so this way we can predict that will keep decreasing towards, meaning that as approaches 0 with smaller values, tends to, and we note: Now let’s evaluate as approaches 0 with values less than zero: We can clearly notice that as gets closer and closer to zero with greater values, the values of the function become bigger and bigger, we can predict then that it keeps increasing in value towards, in other terms as approaches 0 with greater values, tend to, and we note:Īnd we read: the limit of the function as approaches 0 with greater values (or as approaches 0 from the right side) is. Let’s start with evaluating as gets closer and closer to zero with greater values: We know that the function h isn’t defined for (because we can’t divide by 0), but let’s try and see the values of as approaches 0 with both greater and less than 0, meaning we evaluate when approaches 0 from the right side (i.e., ), and when approaches 0 from the left side (i.e., ). Let’s consider the function defined as follow: Here is the graph representing the function, where we can see the graph approaching the X-axis as goes towards infinity. So, From the two previous examples, we conclude that a limit of a function can a real number or.
We can easily notice that as increases in value the value of goes closer and closer to 0, so we can predict that as approaches infinity the value of the function approaches 0, and we note: (To close to zero that the calculator shows exactly 0) Let’s evaluate the function as takes bigger and bigger values:
Here is the graph of the function for a better illustration: In this case, we can use the notation:Īnd we read the limit of the function as approaches, is. We notice that, as the value of increase, the value of the increases as well, so we can predict that as the value of become bigger and bigger, the value of the function goes bigger and bigger too, meaning that as approaches infinity (infinitely big) the value of approaches as well. Let’s evaluate the function when the values of go bigger and bigger To better introduce the idea of limits, let’s take a look at some examples:Įxample 1: Let’s the function be defined as follow: We saw in a previous article, an introduction to functions and some of their properties, in this blog post we will learn about function’s limits, where we will try to the behavior of the function near infinity and near specific real values (i.e., the values for who the function isn’t defined), in other terms, we try to determine the function value when approaching the extremities of its domain of definition. Finite and infinite limits when approaching a real number.Finite and infinite limits when approaching infinity.In this article, we will introduce the idea of limits and the different cases that we can come across. In order to study a function and its behavior and properties, an important step is to find out the limits of the function on the ends of its domain of definition.