One-Sided Limits
Sometimes functions are not continuous, which means they tell us two different limits while approaching from two different sides. For example, the greatest integer function is inherently discontinuous, and at these discontinuities, we observe two different values.
f(x) =[x]
At x = 1, there are two values when approached from two different sides. In such cases, mostly we are asked to tell the value from one side. These are called one-sided limits.
Estimating Limits from Graphs
The concept of limits has been around for thousands of years. Earlier mathematicians in ancient civilizations used limits to approximate the area of a circle. But the formal concept was not around till the 19th century. This concept is essential to calculus and serves as a building block for analyzing derivatives, continuity, and differentiability. Intuitively, limits give us an idea about the values function approaches at a particular value of x. Using this idea, limits can also be estimated to a certain extent just by looking at the graph. Let’s look at these ideas in detail.
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